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Amichay G, Li L, Nagy M, Couzin ID. Revealing the mechanism and function underlying pairwise temporal coupling in collective motion. Nat Commun 2024; 15:4356. [PMID: 38778073 PMCID: PMC11111445 DOI: 10.1038/s41467-024-48458-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2023] [Accepted: 04/29/2024] [Indexed: 05/25/2024] Open
Abstract
Coordinated motion in animal groups has predominantly been studied with a focus on spatial interactions, such as how individuals position and orient themselves relative to one another. Temporal aspects have, by contrast, received much less attention. Here, by studying pairwise interactions in juvenile zebrafish (Danio rerio)-including using immersive volumetric virtual reality (VR) with which we can directly test models of social interactions in situ-we reveal that there exists a rhythmic out-of-phase (i.e., an alternating) temporal coordination dynamic. We find that reciprocal (bi-directional) feedback is both necessary and sufficient to explain this emergent coupling. Beyond a mechanistic understanding, we find, both from VR experiments and analysis of freely swimming pairs, that temporal coordination considerably improves spatial responsiveness, such as to changes in the direction of motion of a partner. Our findings highlight the synergistic role of spatial and temporal coupling in facilitating effective communication between individuals on the move.
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Affiliation(s)
- Guy Amichay
- Centre for the Advanced Study of Collective Behaviour, University of Konstanz, Universitätsstraße 10, 78464, Konstanz, Germany.
- Department of Collective Behaviour, Max-Planck Institute of Animal Behavior, Konstanz, Germany.
- Department of Biology, University of Konstanz, Konstanz, Germany.
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, USA.
- Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL, USA.
| | - Liang Li
- Centre for the Advanced Study of Collective Behaviour, University of Konstanz, Universitätsstraße 10, 78464, Konstanz, Germany
- Department of Collective Behaviour, Max-Planck Institute of Animal Behavior, Konstanz, Germany
- Department of Biology, University of Konstanz, Konstanz, Germany
| | - Máté Nagy
- Centre for the Advanced Study of Collective Behaviour, University of Konstanz, Universitätsstraße 10, 78464, Konstanz, Germany.
- Department of Collective Behaviour, Max-Planck Institute of Animal Behavior, Konstanz, Germany.
- Department of Biology, University of Konstanz, Konstanz, Germany.
- MTA-ELTE Lendület Collective Behaviour Research Group, Hungarian Academy of Sciences, Budapest, Hungary.
- ELTE Eötvös Loránd University, Department of Biological Physics, Budapest, Hungary.
| | - Iain D Couzin
- Centre for the Advanced Study of Collective Behaviour, University of Konstanz, Universitätsstraße 10, 78464, Konstanz, Germany.
- Department of Collective Behaviour, Max-Planck Institute of Animal Behavior, Konstanz, Germany.
- Department of Biology, University of Konstanz, Konstanz, Germany.
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2
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Kanigowski D, Urban-Ciecko J. Conditioning and pseudoconditioning differently change intrinsic excitability of inhibitory interneurons in the neocortex. Cereb Cortex 2024; 34:bhae109. [PMID: 38572735 PMCID: PMC10993172 DOI: 10.1093/cercor/bhae109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2023] [Revised: 02/26/2024] [Accepted: 02/27/2024] [Indexed: 04/05/2024] Open
Abstract
Many studies indicate a broad role of various classes of GABAergic interneurons in the processes related to learning. However, little is known about how the learning process affects intrinsic excitability of specific classes of interneurons in the neocortex. To determine this, we employed a simple model of conditional learning in mice where vibrissae stimulation was used as a conditioned stimulus and a tail shock as an unconditioned one. In vitro whole-cell patch-clamp recordings showed an increase in intrinsic excitability of low-threshold spiking somatostatin-expressing interneurons (SST-INs) in layer 4 (L4) of the somatosensory (barrel) cortex after the conditioning paradigm. In contrast, pseudoconditioning reduced intrinsic excitability of SST-LTS, parvalbumin-expressing interneurons (PV-INs), and vasoactive intestinal polypeptide-expressing interneurons (VIP-INs) with accommodating pattern in L4 of the barrel cortex. In general, increased intrinsic excitability was accompanied by narrowing of action potentials (APs), whereas decreased intrinsic excitability coincided with AP broadening. Altogether, these results show that both conditioning and pseudoconditioning lead to plastic changes in intrinsic excitability of GABAergic interneurons in a cell-specific manner. In this way, changes in intrinsic excitability can be perceived as a common mechanism of learning-induced plasticity in the GABAergic system.
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Affiliation(s)
- Dominik Kanigowski
- Laboratory of Electrophysiology, Nencki Institute of Experimental Biology PAS, 3 Pasteur Street, 02-093 Warsaw, Poland
| | - Joanna Urban-Ciecko
- Laboratory of Electrophysiology, Nencki Institute of Experimental Biology PAS, 3 Pasteur Street, 02-093 Warsaw, Poland
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3
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Dumont G, Pérez-Cervera A, Gutkin B. A framework for macroscopic phase-resetting curves for generalised spiking neural networks. PLoS Comput Biol 2022; 18:e1010363. [PMID: 35913991 PMCID: PMC9371324 DOI: 10.1371/journal.pcbi.1010363] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2021] [Revised: 08/11/2022] [Accepted: 07/06/2022] [Indexed: 11/18/2022] Open
Abstract
Brain rhythms emerge from synchronization among interconnected spiking neurons. Key properties of such rhythms can be gleaned from the phase-resetting curve (PRC). Inferring the PRC and developing a systematic phase reduction theory for large-scale brain rhythms remains an outstanding challenge. Here we present a theoretical framework and methodology to compute the PRC of generic spiking networks with emergent collective oscillations. We adopt a renewal approach where neurons are described by the time since their last action potential, a description that can reproduce the dynamical feature of many cell types. For a sufficiently large number of neurons, the network dynamics are well captured by a continuity equation known as the refractory density equation. We develop an adjoint method for this equation giving a semi-analytical expression of the infinitesimal PRC. We confirm the validity of our framework for specific examples of neural networks. Our theoretical framework can link key biological properties at the individual neuron scale and the macroscopic oscillatory network properties. Beyond spiking networks, the approach is applicable to a broad class of systems that can be described by renewal processes. The formation of oscillatory neuronal assemblies at the network level has been hypothesized to be fundamental to many cognitive and motor functions. One prominent tool to understand the dynamics of oscillatory activity response to stimuli, and hence the neural code for which it is a substrate, is a nonlinear measure called Phase-Resetting Curve (PRC). At the network scale, the PRC defines the measure of how a given synaptic input perturbs the timing of next upcoming volley of spike assemblies: either advancing or delaying this timing. As a further application, one can use PRCs to make unambiguous predictions about whether communicating networks of neurons will phase-lock as it is often observed across the cortical areas and what would be this stable phase-configuration: synchronous, asynchronous or with asymmetric phase-shifts. The latter configuration also implies a preferential flow of information form the leading network to the follower, thereby giving causal signatures of directed functional connectivity. Because of the key position of the PRC in studying synchrony, information flow and entrainment to external forcing, it is crucial to move toward a theory that allows to compute the PRCs of network-wide oscillations not only for a restricted class of models, as has been done in the past, but to network descriptions that are generalized and can reflect flexibly single cell properties. In this manuscript, we tackle this issue by showing how the PRC for network oscillations can be computed using the adjoint systems of partial differential equations that define the dynamics of the neural activity density.
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Affiliation(s)
- Grégory Dumont
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure - PSL University, Paris France
- * E-mail:
| | - Alberto Pérez-Cervera
- Center for Cognition and Decision Making, Institute for Cognitive Neuroscience, National Research University Higher School of Economics, Moscow
- Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Madrid, Spain
| | - Boris Gutkin
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure - PSL University, Paris France
- Center for Cognition and Decision Making, Institute for Cognitive Neuroscience, National Research University Higher School of Economics, Moscow
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4
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Hesse J, Schleimer JH, Maier N, Schmitz D, Schreiber S. Temperature elevations can induce switches to homoclinic action potentials that alter neural encoding and synchronization. Nat Commun 2022; 13:3934. [PMID: 35803913 PMCID: PMC9270341 DOI: 10.1038/s41467-022-31195-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2020] [Accepted: 06/07/2022] [Indexed: 11/09/2022] Open
Abstract
Almost seventy years after the discovery of the mechanisms of action potential generation, some aspects of their computational consequences are still not fully understood. Based on mathematical modeling, we here explore a type of action potential dynamics – arising from a saddle-node homoclinic orbit bifurcation - that so far has received little attention. We show that this type of dynamics is to be expected by specific changes in common physiological parameters, like an elevation of temperature. Moreover, we demonstrate that it favours synchronization patterns in networks – a feature that becomes particularly prominent when system parameters change such that homoclinic spiking is induced. Supported by in-vitro hallmarks for homoclinic spikes in the rodent brain, we hypothesize that the prevalence of homoclinic spikes in the brain may be underestimated and provide a missing link between the impact of biophysical parameters on abrupt transitions between asynchronous and synchronous states of electrical activity in the brain. The intrinsic dynamics of neurons, in particular the generation action potentials, can impact neural network states and processes of encoding information. The authors demonstrate how the elevation of temperature induces a type of action potential dynamics that favors synchronization patterns in neural networks.
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Affiliation(s)
- Janina Hesse
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany.,Bernstein Center for Computational Neuroscience, Berlin, Germany.,Institute for Systems Medicine, Department of Human Medicine, MSH Medical School Hamburg-University of Applied Sciences and Medical University, Hamburg, 20457, Germany.,Charité-Universitätsmedizin Berlin, Corporate member of Freie Universität Berlin, Humboldt-Universität zu Berlin, and Berlin Institute of Health, Berlin, Germany
| | - Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany.,Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Nikolaus Maier
- Neuroscience Research Center - Charité-Universitätsmedizin Berlin, Corporate member of Freie Universität Berlin, Humboldt-Universität zu Berlin, and Berlin Institute of Health, German Center for Neurodegenerative Diseases (DZNE) Berlin, Berlin, 10117, Germany.,Max-Delbrück-Centrum (MDC) for Molecular Medicine, Berlin, 13125, Germany
| | - Dietmar Schmitz
- Bernstein Center for Computational Neuroscience, Berlin, Germany.,Neuroscience Research Center - Charité-Universitätsmedizin Berlin, Corporate member of Freie Universität Berlin, Humboldt-Universität zu Berlin, and Berlin Institute of Health, German Center for Neurodegenerative Diseases (DZNE) Berlin, Berlin, 10117, Germany.,Max-Delbrück-Centrum (MDC) for Molecular Medicine, Berlin, 13125, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany. .,Bernstein Center for Computational Neuroscience, Berlin, Germany.
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5
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Singh MF, Cole MW, Braver TS, Ching S. Developing control-theoretic objectives for large-scale brain dynamics and cognitive enhancement. ANNUAL REVIEWS IN CONTROL 2022; 54:363-376. [PMID: 38250171 PMCID: PMC10798814 DOI: 10.1016/j.arcontrol.2022.05.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/23/2024]
Abstract
The development of technologies for brain stimulation provides a means for scientists and clinicians to directly actuate the brain and nervous system. Brain stimulation has shown intriguing potential in terms of modifying particular symptom clusters in patients and behavioral characteristics of subjects. The stage is thus set for optimization of these techniques and the pursuit of more nuanced stimulation objectives, including the modification of complex cognitive functions such as memory and attention. Control theory and engineering will play a key role in the development of these methods, guiding computational and algorithmic strategies for stimulation. In particular, realizing this goal will require new development of frameworks that allow for controlling not only brain activity, but also latent dynamics that underlie neural computation and information processing. In the current opinion, we review recent progress in brain stimulation and outline challenges and potential research pathways associated with exogenous control of cognitive function.
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Affiliation(s)
- Matthew F Singh
- Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, 63130, MO, USA
- Center for Molecular and Behavioral Neuroscience, Rutgers University, Newark, 07102, NJ, USA
- Psychological and Brain Science, Washington University in St. Louis, St. Louis, 63130, MO, USA
| | - Michael W Cole
- Center for Molecular and Behavioral Neuroscience, Rutgers University, Newark, 07102, NJ, USA
| | - Todd S Braver
- Psychological and Brain Science, Washington University in St. Louis, St. Louis, 63130, MO, USA
| | - ShiNung Ching
- Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, 63130, MO, USA
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6
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Ji H, Fouad AD, Teng S, Liu A, Alvarez-Illera P, Yao B, Li Z, Fang-Yen C. Phase response analyses support a relaxation oscillator model of locomotor rhythm generation in Caenorhabditis elegans. eLife 2021; 10:e69905. [PMID: 34569934 PMCID: PMC8560089 DOI: 10.7554/elife.69905] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2021] [Accepted: 09/24/2021] [Indexed: 01/25/2023] Open
Abstract
Neural circuits coordinate with muscles and sensory feedback to generate motor behaviors appropriate to an animal's environment. In C. elegans, the mechanisms by which the motor circuit generates undulations and modulates them based on the environment are largely unclear. We quantitatively analyzed C. elegans locomotion during free movement and during transient optogenetic muscle inhibition. Undulatory movements were highly asymmetrical with respect to the duration of bending and unbending during each cycle. Phase response curves induced by brief optogenetic inhibition of head muscles showed gradual increases and rapid decreases as a function of phase at which the perturbation was applied. A relaxation oscillator model based on proprioceptive thresholds that switch the active muscle moment was developed and is shown to quantitatively agree with data from free movement, phase responses, and previous results for gait adaptation to mechanical loadings. Our results suggest a neuromuscular mechanism underlying C. elegans motor pattern generation within a compact circuit.
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Affiliation(s)
- Hongfei Ji
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Anthony D Fouad
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Shelly Teng
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Alice Liu
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Pilar Alvarez-Illera
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Bowen Yao
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Zihao Li
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Christopher Fang-Yen
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
- Department of Neuroscience, Perelman School of Medicine, University of PennsylvaniaPhiladelphiaUnited States
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7
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Skilling QM, Eniwaye B, Clawson BC, Shaver J, Ognjanovski N, Aton SJ, Zochowski M. Acetylcholine-gated current translates wake neuronal firing rate information into a spike timing-based code in Non-REM sleep, stabilizing neural network dynamics during memory consolidation. PLoS Comput Biol 2021; 17:e1009424. [PMID: 34543284 PMCID: PMC8483332 DOI: 10.1371/journal.pcbi.1009424] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2021] [Revised: 09/30/2021] [Accepted: 09/06/2021] [Indexed: 11/19/2022] Open
Abstract
Sleep is critical for memory consolidation, although the exact mechanisms mediating this process are unknown. Combining reduced network models and analysis of in vivo recordings, we tested the hypothesis that neuromodulatory changes in acetylcholine (ACh) levels during non-rapid eye movement (NREM) sleep mediate stabilization of network-wide firing patterns, with temporal order of neurons’ firing dependent on their mean firing rate during wake. In both reduced models and in vivo recordings from mouse hippocampus, we find that the relative order of firing among neurons during NREM sleep reflects their relative firing rates during prior wake. Our modeling results show that this remapping of wake-associated, firing frequency-based representations is based on NREM-associated changes in neuronal excitability mediated by ACh-gated potassium current. We also show that learning-dependent reordering of sequential firing during NREM sleep, together with spike timing-dependent plasticity (STDP), reconfigures neuronal firing rates across the network. This rescaling of firing rates has been reported in multiple brain circuits across periods of sleep. Our model and experimental data both suggest that this effect is amplified in neural circuits following learning. Together our data suggest that sleep may bias neural networks from firing rate-based towards phase-based information encoding to consolidate memories. We show that neuromodulatory changes during non-rapid eye movement (NREM) sleep generate stable spike timing relationships between neurons, the ordering of which reflects the neurons’ relative firing rates during wake. Learning-dependent ordering of firing in the hippocampus during NREM, acting in tandem with spike timing-dependent plasticity, reconfigures neuronal firing rates across the hippocampal network. This “rescaling” of neuronal firing rates has recently been reported in multiple brain circuits across periods of sleep. Together, our results suggest that the brain is remapping frequency-biased representations of information formed during wake into timing biased-representations during NREM sleep.
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Affiliation(s)
- Quinton M Skilling
- Biophysics Program, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Bolaji Eniwaye
- Applied Physics Program, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Brittany C Clawson
- Department of Molecular, Cellular, and Developmental Biology, University of Michigan, Ann Arbor, Michigan, United States of America
| | - James Shaver
- Department of Molecular, Cellular, and Developmental Biology, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Nicolette Ognjanovski
- Department of Molecular, Cellular, and Developmental Biology, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Sara J Aton
- Department of Molecular, Cellular, and Developmental Biology, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Michal Zochowski
- Biophysics Program, University of Michigan, Ann Arbor, Michigan, United States of America
- Department of Physics, University of Michigan, Ann Arbor, Michigan, United States of America
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8
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Ramlow L, Lindner B. Interspike interval correlations in neuron models with adaptation and correlated noise. PLoS Comput Biol 2021; 17:e1009261. [PMID: 34449771 PMCID: PMC8428727 DOI: 10.1371/journal.pcbi.1009261] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2020] [Revised: 09/09/2021] [Accepted: 07/08/2021] [Indexed: 11/19/2022] Open
Abstract
The generation of neural action potentials (spikes) is random but nevertheless may result in a rich statistical structure of the spike sequence. In particular, contrary to the popular renewal assumption of theoreticians, the intervals between adjacent spikes are often correlated. Experimentally, different patterns of interspike-interval correlations have been observed and computational studies have identified spike-frequency adaptation and correlated noise as the two main mechanisms that can lead to such correlations. Analytical studies have focused on the single cases of either correlated (colored) noise or adaptation currents in combination with uncorrelated (white) noise. For low-pass filtered noise or adaptation, the serial correlation coefficient can be approximated as a single geometric sequence of the lag between the intervals, providing an explanation for some of the experimentally observed patterns. Here we address the problem of interval correlations for a widely used class of models, multidimensional integrate-and-fire neurons subject to a combination of colored and white noise sources and a spike-triggered adaptation current. Assuming weak noise, we derive a simple formula for the serial correlation coefficient, a sum of two geometric sequences, which accounts for a large class of correlation patterns. The theory is confirmed by means of numerical simulations in a number of special cases including the leaky, quadratic, and generalized integrate-and-fire models with colored noise and spike-frequency adaptation. Furthermore we study the case in which the adaptation current and the colored noise share the same time scale, corresponding to a slow stochastic population of adaptation channels; we demonstrate that our theory can account for a nonmonotonic dependence of the correlation coefficient on the channel’s time scale. Another application of the theory is a neuron driven by network-noise-like fluctuations (green noise). We also discuss the range of validity of our weak-noise theory and show that by changing the relative strength of white and colored noise sources, we can change the sign of the correlation coefficient. Finally, we apply our theory to a conductance-based model which demonstrates its broad applicability. The elementary processing units in the central nervous system are neurons that transmit information by short electrical pulses, so called action potentials or spikes. The generation of the action potential is a random process that can be shaped by correlated fluctuations (colored noise) and by adaptation. A consequence of these two ubiquitous features is that the successive time intervals between spikes, the interspike intervals, are not independent but correlated. As these correlations can significantly improve information transmission and weak-signal detection, it is an important task to develop analytical approaches to these statistics for well-established computational models. Here we present a theory of interval correlations for a widely used class of integrate-and-fire models endowed with an adaptation mechanism and subject to correlated fluctuations. We demonstrate which patterns of interval correlations can be expected from the interplay of colored noise, adaptation and intrinsic nonlinear dynamics.
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Affiliation(s)
- Lukas Ramlow
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
- Physics Department, Humboldt University zu Berlin, Berlin, Germany
- * E-mail:
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
- Physics Department, Humboldt University zu Berlin, Berlin, Germany
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9
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Gast R, Gong R, Schmidt H, Meijer HGE, Knösche TR. On the Role of Arkypallidal and Prototypical Neurons for Phase Transitions in the External Pallidum. J Neurosci 2021; 41:6673-6683. [PMID: 34193559 PMCID: PMC8336705 DOI: 10.1523/jneurosci.0094-21.2021] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2021] [Revised: 04/08/2021] [Accepted: 05/13/2021] [Indexed: 01/10/2023] Open
Abstract
The external pallidum (globus pallidus pars externa [GPe]) plays a central role for basal ganglia functions and dynamics and, consequently, has been included in most computational studies of the basal ganglia. These studies considered the GPe as a homogeneous neural population. However, experimental studies have shown that the GPe contains at least two distinct cell types (prototypical and arkypallidal cells). In this work, we provide in silico insight into how pallidal heterogeneity modulates dynamic regimes inside the GPe and how they affect the GPe response to oscillatory input. We derive a mean-field model of the GPe system from a microscopic spiking neural network of recurrently coupled prototypical and arkypallidal neurons. Using bifurcation analysis, we examine the influence of dopamine-dependent changes of intrapallidal connectivity on the GPe dynamics. We find that increased self-inhibition of prototypical cells can induce oscillations, whereas increased inhibition of prototypical cells by arkypallidal cells leads to the emergence of a bistable regime. Furthermore, we show that oscillatory input to the GPe, arriving from striatum, leads to characteristic patterns of cross-frequency coupling observed at the GPe. Based on these findings, we propose two different hypotheses of how dopamine depletion at the GPe may lead to phase-amplitude coupling between the parkinsonian beta rhythm and a GPe-intrinsic γ rhythm. Finally, we show that these findings generalize to realistic spiking neural networks of sparsely coupled Type I excitable GPe neurons.SIGNIFICANCE STATEMENT Our work provides (1) insight into the theoretical implications of a dichotomous globus pallidus pars externa (GPe) organization, and (2) an exact mean-field model that allows for future investigations of the relationship between GPe spiking activity and local field potential fluctuations. We identify the major phase transitions that the GPe can undergo when subject to static or periodic input and link these phase transitions to the emergence of synchronized oscillations and cross-frequency coupling in the basal ganglia. Because of the close links between our model and experimental findings on the structure and dynamics of prototypical and arkypallidal cells, our results can be used to guide both experimental and computational studies on the role of the GPe for basal ganglia dynamics in health and disease.
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Affiliation(s)
- Richard Gast
- Max Planck Institute for Human Cognitive and Brain Sciences, Brain Networks Group, Leipzig, Germany 04103
| | - Ruxue Gong
- Max Planck Institute for Human Cognitive and Brain Sciences, Brain Networks Group, Leipzig, Germany 04103
| | - Helmut Schmidt
- Max Planck Institute for Human Cognitive and Brain Sciences, Brain Networks Group, Leipzig, Germany 04103
| | - Hil G E Meijer
- Department of Applied Mathematics, Technical Medical Centre, University of Twente, Enschede, The Netherlands 7522 NB
| | - Thomas R Knösche
- Max Planck Institute for Human Cognitive and Brain Sciences, Brain Networks Group, Leipzig, Germany 04103
- Institute for Biomedical Engineering and Informatics, Ilmenau, Germany 98684
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10
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Zeldenrust F, Gutkin B, Denéve S. Efficient and robust coding in heterogeneous recurrent networks. PLoS Comput Biol 2021; 17:e1008673. [PMID: 33930016 PMCID: PMC8115785 DOI: 10.1371/journal.pcbi.1008673] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2021] [Revised: 05/12/2021] [Accepted: 04/07/2021] [Indexed: 11/19/2022] Open
Abstract
Cortical networks show a large heterogeneity of neuronal properties. However, traditional coding models have focused on homogeneous populations of excitatory and inhibitory neurons. Here, we analytically derive a class of recurrent networks of spiking neurons that close to optimally track a continuously varying input online, based on two assumptions: 1) every spike is decoded linearly and 2) the network aims to reduce the mean-squared error between the input and the estimate. From this we derive a class of predictive coding networks, that unifies encoding and decoding and in which we can investigate the difference between homogeneous networks and heterogeneous networks, in which each neurons represents different features and has different spike-generating properties. We find that in this framework, 'type 1' and 'type 2' neurons arise naturally and networks consisting of a heterogeneous population of different neuron types are both more efficient and more robust against correlated noise. We make two experimental predictions: 1) we predict that integrators show strong correlations with other integrators and resonators are correlated with resonators, whereas the correlations are much weaker between neurons with different coding properties and 2) that 'type 2' neurons are more coherent with the overall network activity than 'type 1' neurons.
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Affiliation(s)
- Fleur Zeldenrust
- Department of Neurophysiology, Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands
| | - Boris Gutkin
- Group for Neural Theory, INSERM U960, Département d’Études Cognitives, École Normal Supérieure PSL University, Paris, France
- Center for Cognition and Decision Making, National Research University Higher School of Economics, Moscow, Russia
| | - Sophie Denéve
- Group for Neural Theory, INSERM U960, Département d’Études Cognitives, École Normal Supérieure PSL University, Paris, France
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11
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Evidence of two modes of spiking evoked in human firing motoneurones by Ia afferent electrical stimulation. Exp Brain Res 2021; 239:719-730. [PMID: 33388907 DOI: 10.1007/s00221-020-05998-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2020] [Accepted: 11/24/2020] [Indexed: 10/22/2022]
Abstract
Neurone firing behaviour is a result of complex interaction between synaptic inputs and cellular intrinsic properties. Intriguing firing behaviour, delayed spiking, was shown in some neurones, in particular, in cat neocortical neurones and rat pyramidal hippocampal neurones. In contrast, the similar spiking mode was not reported for animal spinal motoneurones. In the present study, an attempt was made to look for possible evidence of delayed spiking in human motoneurones firing within the low-frequency, sub-primary range, characteristic for voluntary muscle contractions and postural tasks. Forty-seven firing motor units (MUs) were analyzed in ten experiments on three muscles (the flexor carpi ulnaris, the tibialis anterior, and the abductor pollicis brevis) in four healthy humans. Single MUs were activated by gentle voluntary muscle contractions. MU peri-stimulus time histograms, durations of inter-spike intervals, and motoneurone excitability changes within a target interspike interval were analyzed. It was found that during testing the firing motoneurone excitability by small, transient excitatory Ia afferent volley, depending firstly on volley timing within a target interspike interval and excitatory volley strength, the same motoneurone displayed either the direct short-latency response (the H-reflex) or the delayed response (with prolonged and variable latency). Thus, the findings, for the first time, provide evidence for a possibility of two modes of spiking in firing motoneurones. Methods of the estimation of delayed responses and their possible functional significance are discussed. It is emphasized that, for understanding of this issue, the integration of data from studies on experimental animals and humans is desirable.
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Gelastopoulos A, Kopell NJ. Interactions of multiple rhythms in a biophysical network of neurons. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:19. [PMID: 33201339 PMCID: PMC7671958 DOI: 10.1186/s13408-020-00096-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/12/2020] [Accepted: 10/31/2020] [Indexed: 06/11/2023]
Abstract
Neural oscillations, including rhythms in the beta1 band (12-20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different from their natural frequency, but very little is known about how such input affects the network's behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs. We demonstrate that a cell has the ability to respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to that of the other. We show that this is a very general phenomenon, independent of the model used. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it.
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Affiliation(s)
- Alexandros Gelastopoulos
- Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, 02215 Boston, MA USA
- Department of Marketing and Management, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark
| | - Nancy J. Kopell
- Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, 02215 Boston, MA USA
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13
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Zang Y, Hong S, De Schutter E. Firing rate-dependent phase responses of Purkinje cells support transient oscillations. eLife 2020; 9:e60692. [PMID: 32895121 PMCID: PMC7478895 DOI: 10.7554/elife.60692] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2020] [Accepted: 08/20/2020] [Indexed: 01/09/2023] Open
Abstract
Both spike rate and timing can transmit information in the brain. Phase response curves (PRCs) quantify how a neuron transforms input to output by spike timing. PRCs exhibit strong firing-rate adaptation, but its mechanism and relevance for network output are poorly understood. Using our Purkinje cell (PC) model, we demonstrate that the rate adaptation is caused by rate-dependent subthreshold membrane potentials efficiently regulating the activation of Na+ channels. Then, we use a realistic PC network model to examine how rate-dependent responses synchronize spikes in the scenario of reciprocal inhibition-caused high-frequency oscillations. The changes in PRC cause oscillations and spike correlations only at high firing rates. The causal role of the PRC is confirmed using a simpler coupled oscillator network model. This mechanism enables transient oscillations between fast-spiking neurons that thereby form PC assemblies. Our work demonstrates that rate adaptation of PRCs can spatio-temporally organize the PC input to cerebellar nuclei.
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Affiliation(s)
- Yunliang Zang
- Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate UniversityOkinawaJapan
| | - Sungho Hong
- Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate UniversityOkinawaJapan
| | - Erik De Schutter
- Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate UniversityOkinawaJapan
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14
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Pérez-Cervera A, M-Seara T, Huguet G. Global phase-amplitude description of oscillatory dynamics via the parameterization method. CHAOS (WOODBURY, N.Y.) 2020; 30:083117. [PMID: 32872842 DOI: 10.1063/5.0010149] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2020] [Accepted: 07/13/2020] [Indexed: 05/25/2023]
Abstract
In this paper, we use the parameterization method to provide a complete description of the dynamics of an n-dimensional oscillator beyond the classical phase reduction. The parameterization method allows us, via efficient algorithms, to obtain a parameterization of the attracting invariant manifold of the limit cycle in terms of the phase-amplitude variables. The method has several advantages. It provides analytically a Fourier-Taylor expansion of the parameterization up to any order, as well as a simplification of the dynamics that allows for a numerical globalization of the manifolds. Thus, one can obtain the local and global isochrons and isostables, including the slow attracting manifold, up to high accuracy, which offer a geometrical portrait of the oscillatory dynamics. Furthermore, it provides straightforwardly the infinitesimal phase and amplitude response functions, that is, the extended infinitesimal phase and amplitude response curves, which monitor the phase and amplitude shifts beyond the asymptotic state. Thus, the methodology presented yields an accurate description of the phase dynamics for perturbations not restricted to the limit cycle but to its attracting invariant manifold. Finally, we explore some strategies to reduce the dimension of the dynamics, including the reduction of the dynamics to the slow stable submanifold. We illustrate our methods by applying them to different three-dimensional single neuron and neural population models in neuroscience.
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Affiliation(s)
- Alberto Pérez-Cervera
- Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal 647, 08028 Barcelona, Spain
| | - Tere M-Seara
- Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal 647, 08028 Barcelona, Spain
| | - Gemma Huguet
- Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal 647, 08028 Barcelona, Spain
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Simmons DV, Higgs MH, Lebby S, Wilson CJ. Indirect pathway control of firing rate and pattern in the substantia nigra pars reticulata. J Neurophysiol 2020; 123:800-814. [PMID: 31940230 DOI: 10.1152/jn.00678.2019] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022] Open
Abstract
Unitary pallido-nigral synaptic currents were measured using optogenetic stimulation, which activated up to three unitary synaptic inputs to each substantia nigra pars reticulata (SNr) cell. Episodic barrages of synaptic conductances were generated based on in vivo firing patterns of globus pallidus pars externa (GPe) cells and applied to SNr cells using conductance clamp. Barrage inputs were compared to continuous step conductances with the same mean. Barrage inputs and steps both slowed SNr neuron firing and produced disinhibition responses seen in peristimulus histograms. Barrages were less effective than steps at producing inhibition and disinhibition responses. Barrages, but not steps, produced irregular firing during the inhibitory response. Phase models of SNr neurons were constructed from their phase-resetting curves. The phase models reproduced the inhibition and disinhibition responses to the same inputs applied to the neurons. The disinhibition response did not require rebound currents but arose from reset of the cells' oscillation. The differences in firing rate and irregularity in response to barrage and step inhibition resulted from the high sensitivity of SNr neurons to inhibition at late phases in their intrinsic oscillation. During step inhibition, cells continued rhythmic firing at a reduced rate. During barrages, brief bouts of intense inhibition stalled the cells' phase evolution late in their cycle, close to firing, and even very brief respites from inhibition rapidly released single action potentials. The SNr cell firing pattern reflected the fine structure of the synaptic barrage from GPe, as well as its onset and offset.NEW & NOTEWORTHY The pallido-nigral pathway connects the striatum to spontaneously active basal ganglia output neurons in the substantia nigra. Each substantia nigra neuron receives powerful inhibitory synaptic connections from a small group of globus pallidus cells and may fire during pauses in pallidal activity. Despite lacking any hyperpolarization-activated rebound currents, they fire quickly to even brief pauses in the pallido-nigral inhibition. The mechanism of their rapid disinhibitory response is explained by features of their phase-resetting curves.
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Affiliation(s)
- DeNard V Simmons
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas
| | - Matthew H Higgs
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas
| | - Sharmon Lebby
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas
| | - Charles J Wilson
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas
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16
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Chartrand T, Goldman MS, Lewis TJ. Synchronization of Electrically Coupled Resonate-and-Fire Neurons. SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2019; 18:1643-1693. [PMID: 33273894 PMCID: PMC7709966 DOI: 10.1137/18m1197412] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Electrical coupling between neurons is broadly present across brain areas and is typically assumed to synchronize network activity. However, intrinsic properties of the coupled cells can complicate this simple picture. Many cell types with electrical coupling show a diversity of post-spike subthreshold fluctuations, often linked to subthreshold resonance, which are transmitted through electrical synapses in addition to action potentials. Using the theory of weakly coupled oscillators, we explore the effect of both subthreshold and spike-mediated coupling on synchrony in small networks of electrically coupled resonate-and-fire neurons, a hybrid neuron model with damped subthreshold oscillations and a range of post-spike voltage dynamics. We calculate the phase response curve using an extension of the adjoint method that accounts for the discontinuous post-spike reset rule. We find that both spikes and subthreshold fluctuations can jointly promote synchronization. The subthreshold contribution is strongest when the voltage exhibits a significant post-spike elevation in voltage, or plateau potential. Additionally, we show that the geometry of trajectories approaching the spiking threshold causes a "reset-induced shear" effect that can oppose synchrony in the presence of network asymmetry, despite having no effect on the phase-locking of symmetrically coupled pairs.
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Affiliation(s)
- Thomas Chartrand
- Graduate Group in Applied Mathematics, University of California-Davis, Davis, CA 95616. Current address: Allen Institute for Brain Science, Seattle, WA
| | - Mark S Goldman
- Center for Neuroscience, Department of Neurobiology, Physiology and Behavior, Department of Ophthalmology and Vision Science, and Graduate Group in Applied Mathematics, University of California-Davis, Davis, CA 95616
| | - Timothy J Lewis
- Department of Mathematics and Graduate Group in Applied Mathematics, University of California-Davis, Davis, CA 95616
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17
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Higgs MH, Wilson CJ. Frequency-dependent entrainment of striatal fast-spiking interneurons. J Neurophysiol 2019; 122:1060-1072. [PMID: 31314645 DOI: 10.1152/jn.00369.2019] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
Abstract
Striatal fast-spiking interneurons (FSIs) fire in variable-length runs of action potentials at 20-200 spikes/s separated by pauses. In vivo, or with fluctuating applied current, both runs and pauses become briefer and more variable. During runs, spikes are entrained specifically to gamma-frequency components of the input fluctuations. We stimulated parvalbumin-expressing striatal FSIs in mouse brain slices with broadband noise currents added to direct current steps and measured spike entrainment across all frequencies. As the constant current level was increased, FSIs produced longer runs and showed sharper frequency tuning, with best entrainment at the stimulus frequency matching their intrarun firing rate. We separated the contributions of previous spikes from that of the fluctuating stimulus, revealing a strong contribution of previous action potentials to gamma-frequency entrainment. In contrast, after subtraction of the effect inherited from the previous spike, the remaining stimulus contribution to spike generation was less sharply tuned, showing a larger contribution of lower frequencies. The frequency specificity of entrainment within a run was reproduced with a phase resetting model based on experimentally measured phase resetting curves of the same FSIs. In the model, broadly tuned phase entrainment for the first spike in a run evolved into sharply tuned gamma entrainment over the next few spikes. The data and modeling results indicate that for FSIs firing in brief runs and pauses firing within runs is entrained by gamma-frequency components of the input, whereas the onset timing of runs may be sensitive to a wider range of stimulus frequency components.NEW & NOTEWORTHY Specific types of neurons entrain their spikes to particular oscillation frequencies in their synaptic input. This entrainment is commonly understood in terms of the subthreshold voltage response, but how this translates to spiking is not clear. We show that in striatal fast-spiking interneurons, entrainment to gamma-frequency input depends on rhythmic spike runs and is explained by the phase resetting curve, whereas run initiation can be triggered by a broad range of input frequencies.
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Affiliation(s)
- Matthew H Higgs
- Department of Biology, The University of Texas at San Antonio, San Antonio, Texas
| | - Charles J Wilson
- Department of Biology, The University of Texas at San Antonio, San Antonio, Texas
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18
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How to correctly quantify neuronal phase-response curves from noisy recordings. J Comput Neurosci 2019; 47:17-30. [PMID: 31231777 DOI: 10.1007/s10827-019-00719-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2019] [Revised: 04/09/2019] [Accepted: 05/07/2019] [Indexed: 10/26/2022]
Abstract
At the level of individual neurons, various coding properties can be inferred from the input-output relationship of a cell. For small inputs, this relation is captured by the phase-response curve (PRC), which measures the effect of a small perturbation on the timing of the subsequent spike. Experimentally, however, an accurate experimental estimation of PRCs is challenging. Despite elaborate measurement efforts, experimental PRC estimates often cannot be related to those from modeling studies. In particular, experimental PRCs rarely resemble the characteristic theoretical PRC expected close to spike initiation, which is indicative of the underlying spike-onset bifurcation. Here, we show for conductance-based model neurons that the correspondence between theoretical and measured phase-response curve is lost when the stimuli used for the estimation are too large. In this case, the derived phase-response curve is distorted beyond recognition and takes on a generic shape that reflects the measurement protocol and masks the spike-onset bifurcation. We discuss how to identify appropriate stimulus strengths for perturbation and noise-stimulation methods, which permit to estimate PRCs that reliably reflect the spike-onset bifurcation - a task that is particularly difficult if a lower bound for the stimulus amplitude is dictated by prominent intrinsic neuronal noise.
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19
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Dumont G, Gutkin B. Macroscopic phase resetting-curves determine oscillatory coherence and signal transfer in inter-coupled neural circuits. PLoS Comput Biol 2019; 15:e1007019. [PMID: 31071085 PMCID: PMC6529019 DOI: 10.1371/journal.pcbi.1007019] [Citation(s) in RCA: 37] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2018] [Revised: 05/21/2019] [Accepted: 04/10/2019] [Indexed: 01/05/2023] Open
Abstract
Macroscopic oscillations of different brain regions show multiple phase relationships that are persistent across time and have been implicated in routing information. While multiple cellular mechanisms influence the network oscillatory dynamics and structure the macroscopic firing motifs, one of the key questions is to identify the biophysical neuronal and synaptic properties that permit such motifs to arise. A second important issue is how the different neural activity coherence states determine the communication between the neural circuits. Here we analyse the emergence of phase-locking within bidirectionally delayed-coupled spiking circuits in which global gamma band oscillations arise from synaptic coupling among largely excitable neurons. We consider both the interneuronal (ING) and the pyramidal-interneuronal (PING) population gamma rhythms and the inter coupling targeting the pyramidal or the inhibitory neurons. Using a mean-field approach together with an exact reduction method, we reduce each spiking network to a low dimensional nonlinear system and derive the macroscopic phase resetting-curves (mPRCs) that determine how the phase of the global oscillation responds to incoming perturbations. This is made possible by the use of the quadratic integrate-and-fire model together with a Lorentzian distribution of the bias current. Depending on the type of gamma (PING vs. ING), we show that incoming excitatory inputs can either speed up the macroscopic oscillation (phase advance; type I PRC) or induce both a phase advance and a delay (type II PRC). From there we determine the structure of macroscopic coherence states (phase-locking) of two weakly synaptically-coupled networks. To do so we derive a phase equation for the coupled system which links the synaptic mechanisms to the coherence states of the system. We show that a synaptic transmission delay is a necessary condition for symmetry breaking, i.e. a non-symmetric phase lag between the macroscopic oscillations. This potentially provides an explanation to the experimentally observed variety of gamma phase-locking modes. Our analysis further shows that symmetry-broken coherence states can lead to a preferred direction of signal transfer between the oscillatory networks where this directionality also depends on the timing of the signal. Hence we suggest a causal theory for oscillatory modulation of functional connectivity between cortical circuits. Large scale brain oscillations emerge from synaptic interactions within neuronal circuits. Over the past years, such macroscopic rhythms have been suggested to play a crucial role in routing the flow of information across cortical regions, resulting in a functional connectome. The underlying mechanism is cortical oscillations that bind together following a well-known motif called phase-locking. While there is significant experimental support for multiple phase-locking modes in the brain, it is still unclear what is the underlying mechanism that permits macroscopic rhythms to phase lock. In the present paper we take up with this issue, and to show that, one can study the emergent macroscopic phase-locking within the mathematical framework of weakly coupled oscillators. We find that under synaptic delays, fully symmetrically coupled networks can display symmetry-broken states of activity, where one network starts to lead in phase the second (also sometimes known as stuttering states). When we analyse how incoming transient signals affect the coupled system, we find that in the symmetry-broken state, the effect depends strongly on which network is targeted (the leader or the follower) as well as the timing of the input. Hence we show how the dynamics of the emergent phase-locked activity imposes a functional directionality on how signals are processed. We thus offer clarification on the synaptic and circuit properties responsible for the emergence of multiple phase-locking patterns and provide support for its functional implication in information transfer.
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Affiliation(s)
- Grégory Dumont
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, Paris, France
- * E-mail: (GD); (BG)
| | - Boris Gutkin
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, Paris, France
- Center for Cognition and Decision Making, Institute for Cognitive Neuroscience, NRU Higher School of Economics, Moscow, Russia
- * E-mail: (GD); (BG)
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Tiroshi L, Goldberg JA. Population dynamics and entrainment of basal ganglia pacemakers are shaped by their dendritic arbors. PLoS Comput Biol 2019; 15:e1006782. [PMID: 30730886 PMCID: PMC6382172 DOI: 10.1371/journal.pcbi.1006782] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2018] [Revised: 02/20/2019] [Accepted: 01/10/2019] [Indexed: 11/30/2022] Open
Abstract
The theory of phase oscillators is an essential tool for understanding population dynamics of pacemaking neurons. GABAergic pacemakers in the substantia nigra pars reticulata (SNr), a main basal ganglia (BG) output nucleus, receive inputs from the direct and indirect pathways at distal and proximal regions of their dendritic arbors, respectively. We combine theory, optogenetic stimulation and electrophysiological experiments in acute brain slices to ask how dendritic properties impact the propensity of the various inputs, arriving at different locations along the dendrite, to recruit or entrain SNr pacemakers. By combining cable theory with sinusoidally-modulated optogenetic activation of either proximal somatodendritic regions or the entire somatodendritic arbor of SNr neurons, we construct an analytical model that accurately fits the empirically measured somatic current response to inputs arising from illuminating the soma and various portions of the dendritic field. We show that the extent of the dendritic tree that is illuminated generates measurable and systematic differences in the pacemaker’s phase response curve (PRC), causing a shift in its peak. Finally, we show that the divergent PRCs correctly predict differences in two major features of the collective dynamics of SNr neurons: the fidelity of population responses to sudden step-like changes in inputs; and the phase latency at which SNr neurons are entrained by rhythmic stimulation, which can occur in the BG under both physiological and pathophysiological conditions. Our novel method generates measurable and physiologically meaningful spatial effects, and provides the first empirical demonstration of how the collective responses of SNr pacemakers are determined by the transmission properties of their dendrites. SNr dendrites may serve to delay distal striatal inputs so that they impinge on the spike initiation zone simultaneously with pallidal and subthalamic inputs in order to guarantee a fair competition between the influence of the monosynaptic direct- and polysynaptic indirect pathways. The substantia nigra pars reticulata (SNr) is a main output nucleus of the basal ganglia (BG), where inputs from the competing direct and indirect pathways converge onto the same neurons. Interestingly, these inputs are differentially distributed with direct and indirect pathway projections arriving at distal and proximal regions of the dendritic arbor, respectively. We employ a novel method combining theory with electrophysiological experiments and optogenetics to study the distinct effects of inputs arriving at different locations along the dendrite. Our approach represents a useful compromise between complexity and reduction in modelling. Our work addresses the question of high fidelity encoding of inputs by networks of neurons in the new context of pacemaking neurons, which are driven to fire by their intrinsic dynamics rather than by a network state. We provide the first empirical demonstration that dendritic delays can introduce latencies in the responses of a population of neurons that are commensurate with synaptic delays, suggesting a new role for SNr dendrites with implications for BG function.
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Affiliation(s)
- Lior Tiroshi
- Department of Medical Neurobiology, Institute of Medical Research Israel–Canada, The Faculty of Medicine, Jerusalem, Israel
- Edmond and Lily Safra Center for Brain Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel
| | - Joshua A. Goldberg
- Department of Medical Neurobiology, Institute of Medical Research Israel–Canada, The Faculty of Medicine, Jerusalem, Israel
- * E-mail:
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21
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Simmons DV, Higgs MH, Lebby S, Wilson CJ. Predicting responses to inhibitory synaptic input in substantia nigra pars reticulata neurons. J Neurophysiol 2018; 120:2679-2693. [PMID: 30207859 DOI: 10.1152/jn.00535.2018] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022] Open
Abstract
The changes in firing probability produced by a synaptic input are usually visualized using the poststimulus time histogram (PSTH). It would be useful if postsynaptic firing patterns could be predicted from patterns of afferent synaptic activation, but attempts to predict the PSTH from synaptic potential waveforms using reasoning based on voltage trajectory and spike threshold have not been successful, especially for inhibitory inputs. We measured PSTHs for substantia nigra pars reticulata (SNr) neurons inhibited by optogenetic stimulation of striato-nigral inputs or by matching artificial inhibitory conductances applied by dynamic clamp. The PSTH was predicted by a model based on each SNr cell's phase-resetting curve (PRC). Optogenetic activation of striato-nigral input or artificial synaptic inhibition produced a PSTH consisting of an initial depression of firing followed by oscillatory increases and decreases repeating at the SNr cell's baseline firing rate. The phase resetting model produced PSTHs closely resembling the cell data, including the primary pause in firing and the oscillation. Key features of the PSTH, including the onset rate and duration of the initial inhibitory phase, and the subsequent increase in firing probability could be explained from the characteristic shape of the SNr cell's PRC. The rate of damping of the late oscillation was explained by the influence of asynchronous phase perturbations producing firing rate jitter and wander. Our results demonstrate the utility of phase-resetting models as a general method for predicting firing in spontaneously active neurons and their value in interpretation of the striato-nigral PSTH. NEW & NOTEWORTHY The coupling of patterned presynaptic input to sequences of postsynaptic firing is a Gordian knot, complicated by the multidimensionality of neuronal state and the diversity of potential initial states. Even so, it is fundamental for even the simplest understanding of network dynamics. We show that a simple phase-resetting model constructed from experimental measurements can explain and predict the sequence of spike rate changes following synaptic inhibition of an oscillating basal ganglia output neuron.
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Affiliation(s)
- D V Simmons
- Department of Biology, University of Texas at San Antonio , San Antonio, Texas
| | - M H Higgs
- Department of Biology, University of Texas at San Antonio , San Antonio, Texas
| | - S Lebby
- Department of Biology, University of Texas at San Antonio , San Antonio, Texas
| | - C J Wilson
- Department of Biology, University of Texas at San Antonio , San Antonio, Texas
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22
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Li S, Zhang G, Wang J, Deng B. Firing regularity control of single neuron based on closed-loop ISI clamp. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2017.09.008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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23
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How linear response shaped models of neural circuits and the quest for alternatives. Curr Opin Neurobiol 2017; 46:234-240. [DOI: 10.1016/j.conb.2017.09.001] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/24/2017] [Accepted: 09/07/2017] [Indexed: 11/23/2022]
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24
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Wilson CJ. Predicting the response of striatal spiny neurons to sinusoidal input. J Neurophysiol 2017; 118:855-873. [PMID: 28490643 DOI: 10.1152/jn.00143.2017] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2017] [Revised: 04/11/2017] [Accepted: 05/05/2017] [Indexed: 11/22/2022] Open
Abstract
Spike-timing effects of small-amplitude sinusoidal currents were measured in mouse striatal spiny neurons firing repetitively. Spike-timing reliability varied with the stimulus frequency. For frequencies near the cell's firing rate, the cells altered firing rate to match the stimulus and became phase locked to it. The stimulus phase of firing during lock depended on the stimulus frequency relative to the cell's unperturbed firing rate. Interspike intervals during sinusoidal stimulation were predicted using an iterative map constructed from the cells' phase-resetting curve. Variability of interspike intervals was reduced by stimulation at all frequencies higher than about half the cell's unperturbed rate, and interspike intervals were accurately predicted by the map. Long sequences of spike times were predicted by iterating on the map. The accuracy of that prediction varied with frequency. Spike time predictability was highest near and during phase lock. The map predicted the phase of firing on the input and its dependence on stimulus frequency. Prediction errors, when they occurred, were of two kinds: unpredicted variation in interspike interval from intrinsic cell noise and accumulation of prediction errors from previous interspike intervals. Each type of prediction error arose from a different mechanism, and their impact was also predicted from the phase model. When two oscillatory input currents were presented simultaneously, striatal neurons responded selectively to only one of them, the one closest in frequency to the cell's unperturbed firing rate. Their spike times encoded the frequency and phase of that single oscillatory input.NEW & NOTEWORTHY During repetitive firing, the timing of action potentials is determined by the interaction between the input and voltage-sensitive currents throughout the interspike interval. This interaction is encapsulated in the neuron's phase-resetting curve. The phase-resetting curve predicted spike timing to small sinusoidal currents over a wide range of stimulus frequencies. Firing patterns were most sensitive to oscillatory components near the cell's own firing rate, even in the presence of noise and other inputs.
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Affiliation(s)
- Charles J Wilson
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas
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25
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Hesse J, Schleimer JH, Schreiber S. Qualitative changes in phase-response curve and synchronization at the saddle-node-loop bifurcation. Phys Rev E 2017; 95:052203. [PMID: 28618541 DOI: 10.1103/physreve.95.052203] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2016] [Indexed: 06/07/2023]
Abstract
Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature. This bifurcation turns out to induce even more drastic changes in synchronization than the BT transition. This result arises from a direct effect of the saddle-node-loop bifurcation on the limit cycle and hence spike dynamics. In contrast, the BT bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking. We specifically demonstrate that the saddle-node-loop bifurcation (i) ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation, and (ii) results in a symmetry breaking of the system's phase-response curve. The latter entails an increase in synchronization range in pulse-coupled oscillators, such as neurons. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators.
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Affiliation(s)
- Janina Hesse
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
| | - Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
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26
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Madadi Asl M, Valizadeh A, Tass PA. Dendritic and Axonal Propagation Delays Determine Emergent Structures of Neuronal Networks with Plastic Synapses. Sci Rep 2017; 7:39682. [PMID: 28045109 PMCID: PMC5206725 DOI: 10.1038/srep39682] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2016] [Accepted: 11/25/2016] [Indexed: 11/09/2022] Open
Abstract
Spike-timing-dependent plasticity (STDP) modifies synaptic strengths based on the relative timing of pre- and postsynaptic spikes. The temporal order of spikes turned out to be crucial. We here take into account how propagation delays, composed of dendritic and axonal delay times, may affect the temporal order of spikes. In a minimal setting, characterized by neglecting dendritic and axonal propagation delays, STDP eliminates bidirectional connections between two coupled neurons and turns them into unidirectional connections. In this paper, however, we show that depending on the dendritic and axonal propagation delays, the temporal order of spikes at the synapses can be different from those in the cell bodies and, consequently, qualitatively different connectivity patterns emerge. In particular, we show that for a system of two coupled oscillatory neurons, bidirectional synapses can be preserved and potentiated. Intriguingly, this finding also translates to large networks of type-II phase oscillators and, hence, crucially impacts on the overall hierarchical connectivity patterns of oscillatory neuronal networks.
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Affiliation(s)
- Mojtaba Madadi Asl
- Institute for Advanced Studies in Basic Sciences (IASBS), Department of Physics, Zanjan, 45195-1159, Iran
| | - Alireza Valizadeh
- Institute for Advanced Studies in Basic Sciences (IASBS), Department of Physics, Zanjan, 45195-1159, Iran.,Institute for Research in Fundamental Sciences (IPM), School of Cognitive Sciences, Tehran, 19395-5746, Iran
| | - Peter A Tass
- Institute of Neuroscience and Medicine - Neuromodulation (INM-7), Research Center Jülich, Jülich, 52425, Germany.,Stanford University, Department of Neurosurgery, Stanford, CA, 94305, USA.,University of Cologne, Department of Neuromodulation, Cologne, 50937, Germany
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27
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Sakuma S, Mizuno-Matsumoto Y, Nishitani Y, Tamura S. Learning Times Required to Identify the Stimulated Position and Shortening of Propagation Path by Hebb’s Rule in Neural Network. AIMS Neurosci 2017. [DOI: 10.3934/neuroscience.2017.4.238] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
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28
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Ha GE, Lee J, Kwak H, Song K, Kwon J, Jung SY, Hong J, Chang GE, Hwang EM, Shin HS, Lee CJ, Cheong E. The Ca 2+-activated chloride channel anoctamin-2 mediates spike-frequency adaptation and regulates sensory transmission in thalamocortical neurons. Nat Commun 2016; 7:13791. [PMID: 27991499 PMCID: PMC5187435 DOI: 10.1038/ncomms13791] [Citation(s) in RCA: 40] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2016] [Accepted: 11/02/2016] [Indexed: 12/22/2022] Open
Abstract
Neuronal firing patterns, which are crucial for determining the nature of encoded information, have been widely studied; however, the molecular identity and cellular mechanisms of spike-frequency adaptation are still not fully understood. Here we show that spike-frequency adaptation in thalamocortical (TC) neurons is mediated by the Ca2+-activated Cl− channel (CACC) anoctamin-2 (ANO2). Knockdown of ANO2 in TC neurons results in significantly reduced spike-frequency adaptation along with increased tonic spiking. Moreover, thalamus-specific knockdown of ANO2 increases visceral pain responses. These results indicate that ANO2 contributes to reductions in spike generation in highly activated TC neurons and thereby restricts persistent information transmission.
Spike-frequency adaptation in thalamocortical (TC) neurons is important for sensory transmission though the underlying mechanisms are not fully understood. Here, the authors identify a role for the calcium-activated chloride channel, ANO2, in mediating TC spiking adaptations and visceral pain response.
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Affiliation(s)
- Go Eun Ha
- Department of Biotechnology, College of Life Science and Biotechnology, Yonsei University, Seoul 03722, Republic of Korea
| | - Jaekwang Lee
- Center for Neural Science, Korea Institute of Science and Technology, Seoul 02792, Republic of Korea
| | - Hankyul Kwak
- Department of Biotechnology, College of Life Science and Biotechnology, Yonsei University, Seoul 03722, Republic of Korea
| | - Kiyeong Song
- Department of Biotechnology, College of Life Science and Biotechnology, Yonsei University, Seoul 03722, Republic of Korea
| | - Jea Kwon
- Center for Neural Science, Korea Institute of Science and Technology, Seoul 02792, Republic of Korea
| | - Soon-Young Jung
- Center for Neural Science, Korea Institute of Science and Technology, Seoul 02792, Republic of Korea
| | - Joohyeon Hong
- Department of Biotechnology, College of Life Science and Biotechnology, Yonsei University, Seoul 03722, Republic of Korea
| | - Gyeong-Eon Chang
- Department of Biotechnology, College of Life Science and Biotechnology, Yonsei University, Seoul 03722, Republic of Korea
| | - Eun Mi Hwang
- Center for Functional Connectomics, Korea Institute of Science and Technology, Seoul 02792, Republic of Korea
| | - Hee-Sup Shin
- Center for Cognition and Sociality, Institute for Basic Science, Daejeon 34141, Republic of Korea
| | - C Justin Lee
- Center for Neural Science, Korea Institute of Science and Technology, Seoul 02792, Republic of Korea
| | - Eunji Cheong
- Department of Biotechnology, College of Life Science and Biotechnology, Yonsei University, Seoul 03722, Republic of Korea
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29
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Morozova EO, Zakharov D, Gutkin BS, Lapish CC, Kuznetsov A. Dopamine Neurons Change the Type of Excitability in Response to Stimuli. PLoS Comput Biol 2016; 12:e1005233. [PMID: 27930673 PMCID: PMC5145155 DOI: 10.1371/journal.pcbi.1005233] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2016] [Accepted: 11/02/2016] [Indexed: 11/18/2022] Open
Abstract
The dynamics of neuronal excitability determine the neuron's response to stimuli, its synchronization and resonance properties and, ultimately, the computations it performs in the brain. We investigated the dynamical mechanisms underlying the excitability type of dopamine (DA) neurons, using a conductance-based biophysical model, and its regulation by intrinsic and synaptic currents. Calibrating the model to reproduce low frequency tonic firing results in N-methyl-D-aspartate (NMDA) excitation balanced by γ-Aminobutyric acid (GABA)-mediated inhibition and leads to type I excitable behavior characterized by a continuous decrease in firing frequency in response to hyperpolarizing currents. Furthermore, we analyzed how excitability type of the DA neuron model is influenced by changes in the intrinsic current composition. A subthreshold sodium current is necessary for a continuous frequency decrease during application of a negative current, and the low-frequency "balanced" state during simultaneous activation of NMDA and GABA receptors. Blocking this current switches the neuron to type II characterized by the abrupt onset of repetitive firing. Enhancing the anomalous rectifier Ih current also switches the excitability to type II. Key characteristics of synaptic conductances that may be observed in vivo also change the type of excitability: a depolarized γ-Aminobutyric acid receptor (GABAR) reversal potential or co-activation of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (AMPARs) leads to an abrupt frequency drop to zero, which is typical for type II excitability. Coactivation of N-methyl-D-aspartate receptors (NMDARs) together with AMPARs and GABARs shifts the type I/II boundary toward more hyperpolarized GABAR reversal potentials. To better understand how altering each of the aforementioned currents leads to changes in excitability profile of DA neuron, we provide a thorough dynamical analysis. Collectively, these results imply that type I excitability in dopamine neurons might be important for low firing rates and fine-tuning basal dopamine levels, while switching excitability to type II during NMDAR and AMPAR activation may facilitate a transient increase in dopamine concentration, as type II neurons are more amenable to synchronization by mutual excitation.
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Affiliation(s)
- Ekaterina O. Morozova
- Department of Physics, Indiana University, Bloomington, Indiana, United States of America
- Department of Mathematical sciences, Indiana University - Purdue University, Indianapolis, Indiana, United States of America
- * E-mail:
| | | | - Boris S. Gutkin
- Group of Neural Theory, INSERM U960 LNC, IEC, Ecole Normale Superieure PSL University, Paris
- Center for Cognition and Decision Making, NRU HSE, Moscow, Russia
| | - Christopher C. Lapish
- Addiction Neuroscience Program, Indiana University - Purdue University, Indianapolis, Indiana, United States of America
| | - Alexey Kuznetsov
- Department of Mathematical sciences, Indiana University - Purdue University, Indianapolis, Indiana, United States of America
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30
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Ashwin P, Coombes S, Nicks R. Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2016; 6:2. [PMID: 26739133 PMCID: PMC4703605 DOI: 10.1186/s13408-015-0033-6] [Citation(s) in RCA: 103] [Impact Index Per Article: 12.9] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2015] [Accepted: 10/30/2015] [Indexed: 05/20/2023]
Abstract
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.
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Affiliation(s)
- Peter Ashwin
- Centre for Systems Dynamics and Control, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, Exeter, EX4 4QF, UK.
| | - Stephen Coombes
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
| | - Rachel Nicks
- School of Mathematics, University of Birmingham, Watson Building, Birmingham, B15 2TT, UK.
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31
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Mofakham S, Fink CG, Booth V, Zochowski MR. Interplay between excitability type and distributions of neuronal connectivity determines neuronal network synchronization. Phys Rev E 2016; 94:042427. [PMID: 27841569 PMCID: PMC5837280 DOI: 10.1103/physreve.94.042427] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2016] [Indexed: 11/07/2022]
Abstract
While the interplay between neuronal excitability properties and global properties of network topology is known to affect network propensity for synchronization, it is not clear how detailed characteristics of these properties affect spatiotemporal pattern formation. Here we study mixed networks, composed of neurons having type I and/or type II phase response curves, with varying distributions of local and random connections and show that not only average network properties, but also the connectivity distribution statistics, significantly affect network synchrony. Namely, we study networks with fixed networkwide properties, but vary the number of random connections that nodes project. We show that varying node excitability (type I vs type II) influences network synchrony most dramatically for systems with long-tailed distributions of the number of random connections per node. This indicates that a cluster of even a few highly rewired cells with a high propensity for synchronization can alter the degree of synchrony in the network as a whole. We show this effect generally on a network of coupled Kuramoto oscillators and investigate the impact of this effect more thoroughly in pulse-coupled networks of biophysical neurons.
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Affiliation(s)
- Sima Mofakham
- Biophysics Program, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Christian G Fink
- Physics and Astronomy Department and Neuroscience Program, Ohio Wesleyan University, Delaware, Ohio 43015, USA
| | - Victoria Booth
- Mathematics Department and Anesthesiology Department, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Michal R Zochowski
- Biophysics Program, University of Michigan, Ann Arbor, Michigan 48109, USA
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
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32
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Stiefel KM, Ermentrout GB. Neurons as oscillators. J Neurophysiol 2016; 116:2950-2960. [PMID: 27683887 DOI: 10.1152/jn.00525.2015] [Citation(s) in RCA: 36] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2015] [Accepted: 09/27/2016] [Indexed: 01/03/2023] Open
Abstract
Regularly spiking neurons can be described as oscillators. In this article we review some of the insights gained from this conceptualization and their relevance for systems neuroscience. First, we explain how a regularly spiking neuron can be viewed as an oscillator and how the phase-response curve (PRC) describes the response of the neuron's spike times to small perturbations. We then discuss the meaning of the PRC for a single neuron's spiking behavior and review the PRCs measured from a variety of neurons in a range of spiking regimes. Next, we show how the PRC can be related to a number of common measures used to quantify neuronal firing, such as the spike-triggered average and the peristimulus histogram. We further show that the response of a neuron to correlated inputs depends on the shape of the PRC. We then explain how the PRC of single neurons can be used to predict neural network behavior. Given the PRC, conduction delays, and the waveform and time course of the synaptic potentials, it is possible to predict neural population behavior such as synchronization. The PRC also allows us to quantify the robustness of the synchronization to heterogeneity and noise. We finally ask how to combine the measured PRCs and the predictions based on PRC to further the understanding of systems neuroscience. As an example, we discuss how the change of the PRC by the neuromodulator acetylcholine could lead to a destabilization of cortical network dynamics. Although all of these studies are grounded in mathematical abstractions that do not strictly hold in biology, they provide good estimates for the emergence of the brain's network activity from the properties of individual neurons. The study of neurons as oscillators can provide testable hypotheses and mechanistic explanations for systems neuroscience.
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Affiliation(s)
| | - G Bard Ermentrout
- Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania
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33
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Watanabe S, Hirono M. Phase-Dependent Modulation of Oscillatory Phase and Synchrony by Long-Lasting Depolarizing Inputs in Central Neurons. eNeuro 2016; 3:ENEURO.0066-16.2016. [PMID: 27785464 PMCID: PMC5069685 DOI: 10.1523/eneuro.0066-16.2016] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2016] [Revised: 09/22/2016] [Accepted: 09/27/2016] [Indexed: 01/11/2023] Open
Abstract
Oscillatory neural activities have been implicated in various types of information processing in the CNS. The procerebral (PC) lobe of the land mollusk Limax valentianus shows an ongoing oscillatory local field potential (LFP). Olfactory input increases both the frequency and spatial synchrony of the LFP oscillation by a nitric oxide (NO)-mediated mechanism, but how NO modulates the activity in a specific manner has been unclear. In the present study, we used electrical stimulation and NO uncaging to systematically analyze the response of the LFP oscillation and found phase-dependent effects on phase shifting and synchrony. The neurons that presumably release NO in the PC lobe preferentially fired at phases in which NO has a synchronizing effect, suggesting that the timing of NO release is regulated to induce a stereotyped response to natural sensory stimuli. The phase-response curve (PRC) describes the timing dependence of responses of an oscillatory system to external input. PRCs are usually constructed by recording the temporal shifts of the neural activity in response to brief electrical pulses. However, NO evokes a long-lasting depolarization persisting for several cycles of oscillation. The phase-response relationship obtained by NO stimulation was approximately the integral of the PRC. A similar relationship was also shown for regular firing of mouse cerebellar Purkinje cells receiving step depolarization, suggesting the generality of the results to oscillatory neural systems with highly distinct properties. These results indicate novel dynamic effects of long-lasting inputs on network oscillation and synchrony, which are based on simple and ubiquitous mechanisms.
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Affiliation(s)
- Satoshi Watanabe
- Department of Bioengineering and Robotics, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
- Graduate School of Pharmaceutical Sciences, University of Tokyo, Tokyo 113-0033, Japan
| | - Moritoshi Hirono
- Graduate School of Brain Science, Doshisha University, Kyoto 610-0394, Japan
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34
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Hyafil A, Giraud AL, Fontolan L, Gutkin B. Neural Cross-Frequency Coupling: Connecting Architectures, Mechanisms, and Functions. Trends Neurosci 2016; 38:725-740. [PMID: 26549886 DOI: 10.1016/j.tins.2015.09.001] [Citation(s) in RCA: 235] [Impact Index Per Article: 29.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2015] [Revised: 08/14/2015] [Accepted: 09/01/2015] [Indexed: 10/22/2022]
Abstract
Neural oscillations are ubiquitously observed in the mammalian brain, but it has proven difficult to tie oscillatory patterns to specific cognitive operations. Notably, the coupling between neural oscillations at different timescales has recently received much attention, both from experimentalists and theoreticians. We review the mechanisms underlying various forms of this cross-frequency coupling. We show that different types of neural oscillators and cross-frequency interactions yield distinct signatures in neural dynamics. Finally, we associate these mechanisms with several putative functions of cross-frequency coupling, including neural representations of multiple environmental items, communication over distant areas, internal clocking of neural processes, and modulation of neural processing based on temporal predictions.
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Affiliation(s)
- Alexandre Hyafil
- Universitat Pompeu Fabra, Theoretical and Computational Neuroscience, Roc Boronat 138, 08018 Barcelona, Spain; Research Unit, Parc Sanitari Sant Joan de Déu and Universitat de Barcelona, Esplugues de Llobregat, Barcelona, Spain.
| | - Anne-Lise Giraud
- Department of Neuroscience, University of Geneva, Campus Biotech, 9 chemin des Mines, 1211 Geneva, Switzerland
| | - Lorenzo Fontolan
- Department of Neuroscience, University of Geneva, Campus Biotech, 9 chemin des Mines, 1211 Geneva, Switzerland
| | - Boris Gutkin
- Group for Neural Theory, Institut National de la Santé et de la Recherche Médicale (INSERM) Unité 960, Département d'Etudes Cognitives, Ecole Normale Supérieure, 29 rue d'Ulm, 75005 Paris, France; Centre for Cognition and Decision Making, National Research University Higher School of Economics, Myasnitskaya Street 20, Moscow 101000, Russia
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35
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Park Y, Ermentrout B. Weakly coupled oscillators in a slowly varying world. J Comput Neurosci 2016; 40:269-81. [DOI: 10.1007/s10827-016-0596-6] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2015] [Revised: 02/17/2016] [Accepted: 02/22/2016] [Indexed: 10/22/2022]
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36
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Saifee TA, Edwards MJ, Kassavetis P, Gilbertson T. Estimation of the phase response curve from Parkinsonian tremor. J Neurophysiol 2015; 115:310-23. [PMID: 26561596 DOI: 10.1152/jn.00216.2015] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2015] [Accepted: 10/29/2015] [Indexed: 11/22/2022] Open
Abstract
Phase response curves (PRCs), characterizing the response of an oscillator to weak external perturbation, have been estimated from a broad range of biological oscillators, including single neurons in vivo. PRC estimates, in turn, provide an intuitive insight into how oscillatory systems become entrained and how they can be desynchronized. Here, we explore the application of PRC theory to the case of Parkinsonian tremor. Initial attempts to establish a causal effect of subthreshold transcranial magnetic stimulation applied to primary motor cortex on the filtered tremor phase were unsuccessful. We explored the possible explanations of this and demonstrate that assumptions made when estimating the PRC in a traditional setting, such as a single neuron, are not arbitrary when applied to the case of tremor PRC estimation. We go on to extract the PRC of Parkinsonian tremor using an iterative method that requires varying the definition of the tremor cycle and estimating the PRC at multiple peristimulus time samples. Justification for this method is supported by estimates of PRC from simulated single neuron data. We provide an approach to estimating confidence limits for tremor PRC and discuss the interpretational caveats introduced by tremor harmonics and the intrinsic variability of the tremor's period.
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Affiliation(s)
- Tabish A Saifee
- Sobell Department of Motor Neuroscience and Movement Disorders, University College London Institute of Neurology, London, United Kingdom
| | - Mark J Edwards
- Sobell Department of Motor Neuroscience and Movement Disorders, University College London Institute of Neurology, London, United Kingdom
| | - Panagiotis Kassavetis
- Sobell Department of Motor Neuroscience and Movement Disorders, University College London Institute of Neurology, London, United Kingdom
| | - Tom Gilbertson
- Division of Neuroscience, Medical Research Institute, University of Dundee, Dundee, United Kingdom; and Department of Neurology, Ninewells Hospital Dundee, Dundee, United Kingdom
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37
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Huaguang G, Zhiguo Z, Bing J, Shenggen C. Dynamics of on-off neural firing patterns and stochastic effects near a sub-critical Hopf bifurcation. PLoS One 2015; 10:e0121028. [PMID: 25867027 PMCID: PMC4395087 DOI: 10.1371/journal.pone.0121028] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2014] [Accepted: 02/07/2015] [Indexed: 11/18/2022] Open
Abstract
On-off firing patterns, in which repetition of clusters of spikes are interspersed with epochs of subthreshold oscillations or quiescent states, have been observed in various nervous systems, but the dynamics of this event remain unclear. Here, we report that on-off firing patterns observed in three experimental models (rat sciatic nerve subject to chronic constrictive injury, rat CA1 pyramidal neuron, and rabbit blood pressure baroreceptor) appeared as an alternation between quiescent state and burst containing multiple period-1 spikes over time. Burst and quiescent state had various durations. The interspike interval (ISI) series of on-off firing pattern was suggested as stochastic using nonlinear prediction and autocorrelation function. The resting state was changed to a period-1 firing pattern via on-off firing pattern as the potassium concentration, static pressure, or depolarization current was changed. During the changing process, the burst duration of on-off firing pattern increased and the duration of the quiescent state decreased. Bistability of a limit cycle corresponding to period-1 firing and a focus corresponding to resting state was simulated near a sub-critical Hopf bifurcation point in the deterministic Morris-Lecar (ML) model. In the stochastic ML model, noise-induced transitions between the coexisting regimes formed an on-off firing pattern, which closely matched that observed in the experiment. In addition, noise-induced exponential change in the escape rate from the focus, and noise-induced coherence resonance were identified. The distinctions between the on-off firing pattern and stochastic firing patterns generated near three other types of bifurcations of equilibrium points, as well as other viewpoints on the dynamics of on-off firing pattern, are discussed. The results not only identify the on-off firing pattern as noise-induced stochastic firing pattern near a sub-critical Hopf bifurcation point, but also offer practical indicators to discriminate bifurcation types and neural excitability types.
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Affiliation(s)
- Gu Huaguang
- School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
- * E-mail:
| | - Zhao Zhiguo
- School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
| | - Jia Bing
- Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China
| | - Chen Shenggen
- School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
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38
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Leone MJ, Schurter BN, Letson B, Booth V, Zochowski M, Fink CG. Synchronization properties of heterogeneous neuronal networks with mixed excitability type. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032813. [PMID: 25871163 PMCID: PMC4899572 DOI: 10.1103/physreve.91.032813] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2014] [Indexed: 06/04/2023]
Abstract
We study the synchronization of neuronal networks with dynamical heterogeneity, showing that network structures with the same propensity for synchronization (as quantified by master stability function analysis) may develop dramatically different synchronization properties when heterogeneity is introduced with respect to neuronal excitability type. Specifically, we investigate networks composed of neurons with different types of phase response curves (PRCs), which characterize how oscillating neurons respond to excitatory perturbations. Neurons exhibiting type 1 PRC respond exclusively with phase advances, while neurons exhibiting type 2 PRC respond with either phase delays or phase advances, depending on when the perturbation occurs. We find that Watts-Strogatz small world networks transition to synchronization gradually as the proportion of type 2 neurons increases, whereas scale-free networks may transition gradually or rapidly, depending upon local correlations between node degree and excitability type. Random placement of type 2 neurons results in gradual transition to synchronization, whereas placement of type 2 neurons as hubs leads to a much more rapid transition, showing that type 2 hub cells easily "hijack" neuronal networks to synchronization. These results underscore the fact that the degree of synchronization observed in neuronal networks is determined by a complex interplay between network structure and the dynamical properties of individual neurons, indicating that efforts to recover structural connectivity from dynamical correlations must in general take both factors into account.
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Affiliation(s)
- Michael J. Leone
- Mathematics Department, New College of Florida, Sarasota, Florida 34243, USA
- Program in Neural Computation, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
| | | | - Benjamin Letson
- Mathematics Department, Ohio Wesleyan University, Delaware, Ohio 43015, USA
- Mathematics Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Victoria Booth
- Mathematics Department and Anesthesiology Department, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Michal Zochowski
- Physics Department and Biophysics Program, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Christian G. Fink
- Physics Department and Neuroscience Program, Ohio Wesleyan University, Delaware, Ohio 43015, USA
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39
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Couto J, Linaro D, De Schutter E, Giugliano M. On the firing rate dependency of the phase response curve of rat Purkinje neurons in vitro. PLoS Comput Biol 2015; 11:e1004112. [PMID: 25775448 PMCID: PMC4361458 DOI: 10.1371/journal.pcbi.1004112] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2014] [Accepted: 01/05/2015] [Indexed: 12/01/2022] Open
Abstract
Synchronous spiking during cerebellar tasks has been observed across Purkinje cells: however, little is known about the intrinsic cellular mechanisms responsible for its initiation, cessation and stability. The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically. Recently, the PRC theory applied to cerebellar Purkinje cells revealed that these behave as phase-independent integrators at low firing rates, and switch to a phase-dependent mode at high rates. Given the implications for computation and information processing in the cerebellum and the possible role of synchrony in the communication with its post-synaptic targets, we further explored the firing rate dependency of the PRC in Purkinje cells. We isolated key factors for the experimental estimation of the PRC and developed a closed-loop approach to reliably compute the PRC across diverse firing rates in the same cell. Our results show unambiguously that the PRC of individual Purkinje cells is firing rate dependent and that it smoothly transitions from phase independent integrator to a phase dependent mode. Using computational models we show that neither channel noise nor a realistic cell morphology are responsible for the rate dependent shift in the phase response curve.
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Affiliation(s)
- João Couto
- Theoretical Neurobiology and Neuroengineering Laboratory, University of Antwerp, Antwerpen, Belgium
- NeuroElectronics Research Flanders, Leuven, Belgium
| | - Daniele Linaro
- Theoretical Neurobiology and Neuroengineering Laboratory, University of Antwerp, Antwerpen, Belgium
- NeuroElectronics Research Flanders, Leuven, Belgium
| | - E De Schutter
- Theoretical Neurobiology and Neuroengineering Laboratory, University of Antwerp, Antwerpen, Belgium
- Computational Neuroscience Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa, Japan
| | - Michele Giugliano
- Theoretical Neurobiology and Neuroengineering Laboratory, University of Antwerp, Antwerpen, Belgium
- NeuroElectronics Research Flanders, Leuven, Belgium
- Department of Computer Science, University of Sheffield, Sheffield, United Kingdom
- Brain Mind Institute, EPFL, Lausanne, Switzerland
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Ratté S, Lankarany M, Rho YA, Patterson A, Prescott SA. Subthreshold membrane currents confer distinct tuning properties that enable neurons to encode the integral or derivative of their input. Front Cell Neurosci 2015; 8:452. [PMID: 25620913 PMCID: PMC4288132 DOI: 10.3389/fncel.2014.00452] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2014] [Accepted: 12/15/2014] [Indexed: 11/25/2022] Open
Abstract
Neurons rely on action potentials, or spikes, to encode information. But spikes can encode different stimulus features in different neurons. We show here through simulations and experiments how neurons encode the integral or derivative of their input based on the distinct tuning properties conferred upon them by subthreshold currents. Slow-activating subthreshold inward (depolarizing) current mediates positive feedback control of subthreshold voltage, sustaining depolarization and allowing the neuron to spike on the basis of its integrated stimulus waveform. Slow-activating subthreshold outward (hyperpolarizing) current mediates negative feedback control of subthreshold voltage, truncating depolarization and forcing the neuron to spike on the basis of its differentiated stimulus waveform. Depending on its direction, slow-activating subthreshold current cooperates or competes with fast-activating inward current during spike initiation. This explanation predicts that sensitivity to the rate of change of stimulus intensity differs qualitatively between integrators and differentiators. This was confirmed experimentally in spinal sensory neurons that naturally behave as specialized integrators or differentiators. Predicted sensitivity to different stimulus features was confirmed by covariance analysis. Integration and differentiation, which are themselves inverse operations, are thus shown to be implemented by the slow feedback mediated by oppositely directed subthreshold currents expressed in different neurons.
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Affiliation(s)
- Stéphanie Ratté
- Neurosciences and Mental Health, The Hospital for Sick Children Toronto, ON, Canada ; Department of Physiology and Institute of Biomaterials and Biomedical Engineering, University of Toronto Toronto, ON, Canada ; Pittsburgh Center for Pain Research, University of Pittsburgh Pittsburgh, PA, USA
| | - Milad Lankarany
- Neurosciences and Mental Health, The Hospital for Sick Children Toronto, ON, Canada ; Department of Physiology and Institute of Biomaterials and Biomedical Engineering, University of Toronto Toronto, ON, Canada
| | - Young-Ah Rho
- Pittsburgh Center for Pain Research, University of Pittsburgh Pittsburgh, PA, USA
| | - Adam Patterson
- Neurosciences and Mental Health, The Hospital for Sick Children Toronto, ON, Canada
| | - Steven A Prescott
- Neurosciences and Mental Health, The Hospital for Sick Children Toronto, ON, Canada ; Department of Physiology and Institute of Biomaterials and Biomedical Engineering, University of Toronto Toronto, ON, Canada ; Pittsburgh Center for Pain Research, University of Pittsburgh Pittsburgh, PA, USA
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Climer JR, DiTullio R, Newman EL, Hasselmo ME, Eden UT. Examination of rhythmicity of extracellularly recorded neurons in the entorhinal cortex. Hippocampus 2014; 25:460-73. [PMID: 25331248 DOI: 10.1002/hipo.22383] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 10/14/2014] [Indexed: 12/16/2022]
Abstract
A number of studies have examined the theta-rhythmic modulation of neuronal firing in the hippocampal circuit. For extracellular recordings, this is often done by examining spectral properties of the spike-time autocorrelogram, most significantly, for validating the presence or absence of theta modulation across species. These techniques can show significant rhythmicity for high firing rate, highly rhythmic neurons; however, they are substantially biased by several factors including the peak firing rate of the neuron, the amount of time spent in the neuron's receptive field, and other temporal properties of the rhythmicity such as cycle-skipping. These limitations make it difficult to examine rhythmic modulation in neurons with low firing rates or when an animal has short dwell times within the firing field and difficult to compare rhythmicity under disparate experimental conditions when these factors frequently differ. Here, we describe in detail the challenges that researchers face when using these techniques and apply our findings to recent recordings from bat entorhinal grid cells, suggesting that they may have lacked enough data to examine theta rhythmicity robustly. We describe a more sensitive and statistically rigorous method using maximum likelihood estimation (MLE) of a parametric model of the lags within the autocorrelation window, which helps to alleviate some of the problems of traditional methods and was also unable to detect rhythmicity in bat grid cells. Using large batteries of simulated data, we explored the boundaries for which the MLE technique and the theta index can detect rhythmicity. The MLE technique is less sensitive to many features of the autocorrelogram and provides a framework for statistical testing to detect rhythmicity as well as changes in rhythmicity in individual sessions providing a substantial improvement over previous methods.
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Affiliation(s)
- Jason R Climer
- Department of Psychological and Brain Sciences, Center for Memory and Brain, Boston University, Massachusetts
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Abstract
We use computational models to examine the circumstances under which modulation of a single neuron can alter the dynamics of an entire circuit. We show that under some circumstances, the circuit's behavior is robust to the neuromodulation of a single hub neuron, while under other circumstances the same neuromodulatory action of a single neuron can produce a large variety of circuit outcomes. Neuromodulation of a single neuron in a circuit can either have little to no effect on the output of the circuit, or it can change the pattern of activity within that circuit. This is dependent on the synaptic parameters given the same circuit architecture, thus illustrating the insufficiency of the connectome alone for determining circuit behavior. ![]()
When does neuromodulation of a single neuron influence the output of the entire network? We constructed a five-cell circuit in which a neuron is at the center of the circuit and the remaining neurons form two distinct oscillatory subnetworks. All neurons were modeled as modified Morris−Lecar models with a hyperpolarization-activated conductance (ḡh) in addition to calcium (ḡCa), potassium (ḡK), and leak conductances. We determined the effects of varying ḡCa, ḡK, and ḡh on the frequency, amplitude, and duty cycle of a single neuron oscillator. The frequency of the single neuron was highest when the ḡK and ḡh conductances were high and ḡCa was moderate whereas, in the traditional Morris−Lecar model, the highest frequencies occur when both ḡK and ḡCa are high. We randomly sampled parameter space to find 143 hub oscillators with nearly identical frequencies but with disparate maximal conductance, duty cycles, and burst amplitudes, and then embedded each of these hub neurons into networks with different sets of synaptic parameters. For one set of network parameters, circuit behavior was virtually identical regardless of the underlying conductances of the hub neuron. For a different set of network parameters, circuit behavior varied with the maximal conductances of the hub neuron. This demonstrates that neuromodulation of a single target neuron may dramatically alter the performance of an entire network when the network is in one state, but have almost no effect when the circuit is in a different state.
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Adaptation and shunting inhibition leads to pyramidal/interneuron gamma with sparse firing of pyramidal cells. J Comput Neurosci 2014; 37:357-76. [PMID: 25005326 DOI: 10.1007/s10827-014-0508-6] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2012] [Revised: 05/29/2014] [Accepted: 06/02/2014] [Indexed: 10/25/2022]
Abstract
Gamma oscillations are a prominent phenomenon related to a number of brain functions. Data show that individual pyramidal neurons can fire at rate below gamma with the population showing clear gamma oscillations and synchrony. In one kind of idealized model of such weak gamma, pyramidal neurons fire in clusters. Here we provide a theory for clustered gamma PING rhythms with strong inhibition and weaker excitation. Our simulations of biophysical models show that the adaptation of pyramidal neurons coupled with their low firing rate leads to cluster formation. A partially analytic study of a canonical model shows that the phase response curves with a near zero flat region, caused by the presence of the slow adaptive current, are the key to the formation of clusters. Furthermore we examine shunting inhibition and show that clusters become robust and generic.
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Predicting the responses of repetitively firing neurons to current noise. PLoS Comput Biol 2014; 10:e1003612. [PMID: 24809636 PMCID: PMC4014400 DOI: 10.1371/journal.pcbi.1003612] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2014] [Accepted: 03/26/2014] [Indexed: 11/22/2022] Open
Abstract
We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically. Most neurons receive thousands of synaptic inputs per second. Each of these may be individually weak but collectively they shape the temporal pattern of firing by the postsynaptic neuron. If the postsynaptic neuron fires repetitively, its synaptic inputs need not directly trigger action potentials, but may instead control the timing of action potentials that would occur anyway. The phase resetting curve encapsulates the influence of an input on the timing of the next action potential, depending on its time of arrival. We measured the phase resetting curves of neurons in the subthalamic nucleus and used them to accurately predict the timing of action potentials in a phase model subjected to complex input patterns. A simple approximation to the phase model accurately predicted the changes in firing pattern evoked by dense patterns of noise pulses varying in amplitude and pulse duration, and by changes in firing rate. We also showed that the phase resetting curve changes systematically with changes in total neuron conductance, and doing so predicts corresponding changes in firing pattern. Our results indicate that the phase model may accurately represent the temporal integration of complex patterns of input to repetitively firing neurons.
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Langfield P, Krauskopf B, Osinga HM. Solving Winfree's puzzle: the isochrons in the FitzHugh-Nagumo model. CHAOS (WOODBURY, N.Y.) 2014; 24:013131. [PMID: 24697393 DOI: 10.1063/1.4867877] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
We consider the FitzHugh-Nagumo model, an example of a system with two time scales for which Winfree was unable to determine the overall structure of the isochrons. An isochron is the set of all points in the basin of an attracting periodic orbit that converge to this periodic orbit with the same asymptotic phase. We compute the isochrons as one-dimensional parametrised curves with a method based on the continuation of suitable two-point boundary value problems. This allows us to present in detail the geometry of how the basin of attraction is foliated by isochrons. They exhibit extreme sensitivity and feature sharp turns, which is why Winfree had difficulties finding them. We observe that the sharp turns and sensitivity of the isochrons are associated with the slow-fast nature of the FitzHugh-Nagumo system; more specifically, it occurs near its repelling (unstable) slow manifold.
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Affiliation(s)
- Peter Langfield
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
| | - Bernd Krauskopf
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
| | - Hinke M Osinga
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
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46
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Dynamics of the exponential integrate-and-fire model with slow currents and adaptation. J Comput Neurosci 2014; 37:161-80. [PMID: 24443127 PMCID: PMC4082791 DOI: 10.1007/s10827-013-0494-0] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2013] [Revised: 12/18/2013] [Accepted: 12/26/2013] [Indexed: 12/03/2022]
Abstract
In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spike-frequency adaptation, phase-response curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in time-periodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIF-type networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPA-like synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HH-type model. Our work shows that the EIF model with additional slow currents is well suited for use in large-scale, point-neuron models in which spike-frequency adaptation is important.
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47
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Canavier CC, Wang S, Chandrasekaran L. Effect of phase response curve skew on synchronization with and without conduction delays. Front Neural Circuits 2013; 7:194. [PMID: 24376399 PMCID: PMC3858834 DOI: 10.3389/fncir.2013.00194] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2013] [Accepted: 11/23/2013] [Indexed: 11/13/2022] Open
Abstract
A central problem in cortical processing including sensory binding and attentional gating is how neurons can synchronize their responses with zero or near-zero time lag. For a spontaneously firing neuron, an input from another neuron can delay or advance the next spike by different amounts depending upon the timing of the input relative to the previous spike. This information constitutes the phase response curve (PRC). We present a simple graphical method for determining the effect of PRC shape on synchronization tendencies and illustrate it using type 1 PRCs, which consist entirely of advances (delays) in response to excitation (inhibition). We obtained the following generic solutions for type 1 PRCs, which include the pulse-coupled leaky integrate and fire model. For pairs with mutual excitation, exact synchrony can be stable for strong coupling because of the stabilizing effect of the causal limit region of the PRC in which an input triggers a spike immediately upon arrival. However, synchrony is unstable for short delays, because delayed inputs arrive during a refractory period and cannot trigger an immediate spike. Right skew destabilizes antiphase and enables modes with time lags that grow as the conduction delay is increased. Therefore, right skew favors near synchrony at short conduction delays and a gradual transition between synchrony and antiphase for pairs coupled by mutual excitation. For pairs with mutual inhibition, zero time lag synchrony is stable for conduction delays ranging from zero to a substantial fraction of the period for pairs. However, for right skew there is a preferred antiphase mode at short delays. In contrast to mutual excitation, left skew destabilizes antiphase for mutual inhibition so that synchrony dominates at short delays as well. These pairwise synchronization tendencies constrain the synchronization properties of neurons embedded in larger networks.
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Affiliation(s)
- Carmen C Canavier
- Department of Cell Biology and Anatomy, Louisiana State University School of Medicine, Louisiana State University Health Sciences Center New Orleans, LA, USA ; Neuroscience Center, Louisiana State University Health Sciences Center New Orleans, LA, USA
| | - Shuoguo Wang
- Department of Cell Biology and Anatomy, Louisiana State University School of Medicine, Louisiana State University Health Sciences Center New Orleans, LA, USA
| | - Lakshmi Chandrasekaran
- Department of Cell Biology and Anatomy, Louisiana State University School of Medicine, Louisiana State University Health Sciences Center New Orleans, LA, USA
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48
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Cash SS. Status epilepticus as a system disturbance: is status epilepticus due to synchronization or desynchronization? Epilepsia 2013; 54 Suppl 6:37-9. [PMID: 24001069 DOI: 10.1111/epi.12273] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The traditional view of seizure activity is one in which there is extreme hypersynchronization. Although what is meant by hypersynchronization is rarely explicitly and fully defined, it can be understood to imply large numbers of neurons firing together essentially simultaneously. In this discussion we explore the possibility that seizures-both self-terminating and sustained in status-are not purely synchronous in time or in space. We investigate the alternative possibility that much seizure activity represents spatiotemporal desynchronization. Furthermore, we discuss the possibility that, in contrast to canonical views of epileptic activity, a high degree of synchronization is a prerequisite for termination of the seizure rather than a marker of early and ongoing seizure activity. These ideas will be discussed with reference to results from our collaborative group based on microelectrode recordings in patients with epilepsy as well as to the many studies done by others in both patients and animal models. Finally, we will explore implications for these hypotheses in the treatment of patients with epilepsy and in status epilepticus.
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Affiliation(s)
- Sydney S Cash
- Department of Neurology, Massachusetts General Hospital and Harvard Medical School, 55 Fruit Street, Boston, MA 02114, U.S.A.
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Oprisan SA. Local linear approximation of the Jacobian matrix better captures phase resetting of neural limit cycle oscillators. Neural Comput 2013; 26:132-57. [PMID: 24102127 DOI: 10.1162/neco_a_00536] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
One effect of any external perturbations, such as presynaptic inputs, received by limit cycle oscillators when they are part of larger neural networks is a transient change in their firing rate, or phase resetting. A brief external perturbation moves the figurative point outside the limit cycle, a geometric perturbation that we mapped into a transient change in the firing rate, or a temporal phase resetting. In order to gain a better qualitative understanding of the link between the geometry of the limit cycle and the phase resetting curve (PRC), we used a moving reference frame with one axis tangent and the others normal to the limit cycle. We found that the stability coefficients associated with the unperturbed limit cycle provided good quantitative predictions of both the tangent and the normal geometric displacements induced by external perturbations. A geometric-to-temporal mapping allowed us to correctly predict the PRC while preserving the intuitive nature of this geometric approach.
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Affiliation(s)
- Sorinel Adrian Oprisan
- Department of Physics and Astronomy, College of Charleston, Charleston, SC 29424, U.S.A.
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Dodla R, Wilson CJ. Interaction function of oscillating coupled neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042704. [PMID: 24229210 PMCID: PMC3928969 DOI: 10.1103/physreve.88.042704] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2013] [Revised: 08/28/2013] [Indexed: 06/02/2023]
Abstract
Large scale simulations of electrically coupled neuronal oscillators often employ the phase coupled oscillator paradigm to understand and predict network behavior. We study the nature of the interaction between such coupled oscillators using weakly coupled oscillator theory. By employing piecewise linear approximations for phase response curves and voltage time courses and parametrizing their shapes, we compute the interaction function for all such possible shapes and express it in terms of discrete Fourier modes. We find that reasonably good approximation is achieved with four Fourier modes that comprise of both sine and cosine terms.
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Affiliation(s)
- Ramana Dodla
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas 78249, USA
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