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Behbood M, Lemaire L, Schleimer JH, Schreiber S. The Na+/K+-ATPase generically enables deterministic bursting in class I neurons by shearing the spike-onset bifurcation structure. PLoS Comput Biol 2024; 20:e1011751. [PMID: 39133755 PMCID: PMC11383233 DOI: 10.1371/journal.pcbi.1011751] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2023] [Revised: 09/09/2024] [Accepted: 08/02/2024] [Indexed: 09/10/2024] Open
Abstract
Slow brain rhythms, for example during slow-wave sleep or pathological conditions like seizures and spreading depolarization, can be accompanied by oscillations in extracellular potassium concentration. Such slow brain rhythms typically have a lower frequency than tonic action-potential firing. They are assumed to arise from network-level mechanisms, involving synaptic interactions and delays, or from intrinsically bursting neurons. Neuronal burst generation is commonly attributed to ion channels with slow kinetics. Here, we explore an alternative mechanism generically available to all neurons with class I excitability. It is based on the interplay of fast-spiking voltage dynamics with a one-dimensional slow dynamics of the extracellular potassium concentration, mediated by the activity of the Na+/K+-ATPase. We use bifurcation analysis of the complete system as well as the slow-fast method to reveal that this coupling suffices to generate a hysteresis loop organized around a bistable region that emerges from a saddle-node loop bifurcation-a common feature of class I excitable neurons. Depending on the strength of the Na+/K+-ATPase, bursts are generated from pump-induced shearing the bifurcation structure, spiking is tonic, or cells are silenced via depolarization block. We suggest that transitions between these dynamics can result from disturbances in extracellular potassium regulation, such as glial malfunction or hypoxia affecting the Na+/K+-ATPase activity. The identified minimal mechanistic model outlining the sodium-potassium pump's generic contribution to burst dynamics can, therefore, contribute to a better mechanistic understanding of pathologies such as epilepsy syndromes and, potentially, inform therapeutic strategies.
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Affiliation(s)
- Mahraz Behbood
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Louisiane Lemaire
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
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2
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Gowers RP, Schreiber S. How neuronal morphology impacts the synchronisation state of neuronal networks. PLoS Comput Biol 2024; 20:e1011874. [PMID: 38437226 PMCID: PMC10939433 DOI: 10.1371/journal.pcbi.1011874] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2023] [Revised: 03/14/2024] [Accepted: 01/30/2024] [Indexed: 03/06/2024] Open
Abstract
The biophysical properties of neurons not only affect how information is processed within cells, they can also impact the dynamical states of the network. Specifically, the cellular dynamics of action-potential generation have shown relevance for setting the (de)synchronisation state of the network. The dynamics of tonically spiking neurons typically fall into one of three qualitatively distinct types that arise from distinct mathematical bifurcations of voltage dynamics at the onset of spiking. Accordingly, changes in ion channel composition or even external factors, like temperature, have been demonstrated to switch network behaviour via changes in the spike onset bifurcation and hence its associated dynamical type. A thus far less addressed modulator of neuronal dynamics is cellular morphology. Based on simplified and anatomically realistic mathematical neuron models, we show here that the extent of dendritic arborisation has an influence on the neuronal dynamical spiking type and therefore on the (de)synchronisation state of the network. Specifically, larger dendritic trees prime neuronal dynamics for in-phase-synchronised or splayed-out activity in weakly coupled networks, in contrast to cells with otherwise identical properties yet smaller dendrites. Our biophysical insights hold for generic multicompartmental classes of spiking neuron models (from ball-and-stick-type to anatomically reconstructed models) and establish a connection between neuronal morphology and the susceptibility of neural tissue to synchronisation in health and disease.
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Affiliation(s)
- Robert P Gowers
- Institute for Theoretical Biology, Humboldt-University of Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Humboldt-University of Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience, Berlin, Germany
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3
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Bernáez Timón L, Ekelmans P, Kraynyukova N, Rose T, Busse L, Tchumatchenko T. How to incorporate biological insights into network models and why it matters. J Physiol 2023; 601:3037-3053. [PMID: 36069408 DOI: 10.1113/jp282755] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2022] [Accepted: 08/24/2022] [Indexed: 11/08/2022] Open
Abstract
Due to the staggering complexity of the brain and its neural circuitry, neuroscientists rely on the analysis of mathematical models to elucidate its function. From Hodgkin and Huxley's detailed description of the action potential in 1952 to today, new theories and increasing computational power have opened up novel avenues to study how neural circuits implement the computations that underlie behaviour. Computational neuroscientists have developed many models of neural circuits that differ in complexity, biological realism or emergent network properties. With recent advances in experimental techniques for detailed anatomical reconstructions or large-scale activity recordings, rich biological data have become more available. The challenge when building network models is to reflect experimental results, either through a high level of detail or by finding an appropriate level of abstraction. Meanwhile, machine learning has facilitated the development of artificial neural networks, which are trained to perform specific tasks. While they have proven successful at achieving task-oriented behaviour, they are often abstract constructs that differ in many features from the physiology of brain circuits. Thus, it is unclear whether the mechanisms underlying computation in biological circuits can be investigated by analysing artificial networks that accomplish the same function but differ in their mechanisms. Here, we argue that building biologically realistic network models is crucial to establishing causal relationships between neurons, synapses, circuits and behaviour. More specifically, we advocate for network models that consider the connectivity structure and the recorded activity dynamics while evaluating task performance.
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Affiliation(s)
- Laura Bernáez Timón
- Institute for Physiological Chemistry, University of Mainz Medical Center, Mainz, Germany
| | - Pierre Ekelmans
- Frankfurt Institute for Advanced Studies, Frankfurt, Germany
| | - Nataliya Kraynyukova
- Institute of Experimental Epileptology and Cognition Research, University of Bonn Medical Center, Bonn, Germany
| | - Tobias Rose
- Institute of Experimental Epileptology and Cognition Research, University of Bonn Medical Center, Bonn, Germany
| | - Laura Busse
- Division of Neurobiology, Faculty of Biology, LMU Munich, Munich, Germany
- Bernstein Center for Computational Neuroscience, Munich, Germany
| | - Tatjana Tchumatchenko
- Institute for Physiological Chemistry, University of Mainz Medical Center, Mainz, Germany
- Institute of Experimental Epileptology and Cognition Research, University of Bonn Medical Center, Bonn, Germany
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4
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Hürkey S, Niemeyer N, Schleimer JH, Ryglewski S, Schreiber S, Duch C. Gap junctions desynchronize a neural circuit to stabilize insect flight. Nature 2023:10.1038/s41586-023-06099-0. [PMID: 37225999 DOI: 10.1038/s41586-023-06099-0] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2022] [Accepted: 04/18/2023] [Indexed: 05/26/2023]
Abstract
Insect asynchronous flight is one of the most prevalent forms of animal locomotion used by more than 600,000 species. Despite profound insights into the motor patterns1, biomechanics2,3 and aerodynamics underlying asynchronous flight4,5, the architecture and function of the central-pattern-generating (CPG) neural network remain unclear. Here, on the basis of an experiment-theory approach including electrophysiology, optophysiology, Drosophila genetics and mathematical modelling, we identify a miniaturized circuit solution with unexpected properties. The CPG network consists of motoneurons interconnected by electrical synapses that, in contrast to doctrine, produce network activity splayed out in time instead of synchronized across neurons. Experimental and mathematical evidence support a generic mechanism for network desynchronization that relies on weak electrical synapses and specific excitability dynamics of the coupled neurons. In small networks, electrical synapses can synchronize or desynchronize network activity, depending on the neuron-intrinsic dynamics and ion channel composition. In the asynchronous flight CPG, this mechanism translates unpatterned premotor input into stereotyped neuronal firing with fixed sequences of cell activation that ensure stable wingbeat power and, as we show, is conserved across multiple species. Our findings prove a wider functional versatility of electrical synapses in the dynamic control of neural circuits and highlight the relevance of detecting electrical synapses in connectomics.
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Affiliation(s)
- Silvan Hürkey
- Institute of Developmental Biology and Neurobiology (iDN), Johannes Gutenberg-University Mainz, Mainz, Germany
| | - Nelson Niemeyer
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany
| | | | - Stefanie Ryglewski
- Institute of Developmental Biology and Neurobiology (iDN), Johannes Gutenberg-University Mainz, Mainz, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany.
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany.
| | - Carsten Duch
- Institute of Developmental Biology and Neurobiology (iDN), Johannes Gutenberg-University Mainz, Mainz, Germany.
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5
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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] [Key Words] [MESH Headings] [Grants] [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.
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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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6
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Pfeiffer P, Barreda Tomás FJ, Wu J, Schleimer JH, Vida I, Schreiber S. A dynamic clamp protocol to artificially modify cell capacitance. eLife 2022; 11:75517. [PMID: 35362411 PMCID: PMC9135398 DOI: 10.7554/elife.75517] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2021] [Accepted: 03/17/2022] [Indexed: 11/13/2022] Open
Abstract
Dynamics of excitable cells and networks depend on the membrane time constant, set by membrane resistance and capacitance. Whereas pharmacological and genetic manipulations of ionic conductances of excitable membranes are routine in electrophysiology, experimental control over capacitance remains a challenge. Here, we present capacitance clamp, an approach that allows electrophysiologists to mimic a modified capacitance in biological neurons via an unconventional application of the dynamic clamp technique. We first demonstrate the feasibility to quantitatively modulate capacitance in a mathematical neuron model and then confirm the functionality of capacitance clamp in in vitro experiments in granule cells of rodent dentate gyrus with up to threefold virtual capacitance changes. Clamping of capacitance thus constitutes a novel technique to probe and decipher mechanisms of neuronal signaling in ways that were so far inaccessible to experimental electrophysiology.
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Affiliation(s)
- Paul Pfeiffer
- Institute for Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
| | | | - Jiameng Wu
- Institute for Integrative Neuroanatomy, Charité - Universitätsmedizin Berlin, Berlin, Germany
| | - Jan-Hendrik Schleimer
- Institute of Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
| | - Imre Vida
- Institute for Integrative Neuroanatomy, Charité - Universitätsmedizin Berlin, Berlin, Germany
| | - Susanne Schreiber
- Institute of Theoretical Biology, Department of Biology, Humboldt-Universität zu Berlin, Berlin, Germany
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7
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Wu Y, Jiao Y, Zhao Y, Jia H, Xu L. Noise-induced quasiperiod and period switching. Phys Rev E 2022; 105:014419. [PMID: 35193235 DOI: 10.1103/physreve.105.014419] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2021] [Accepted: 01/03/2022] [Indexed: 06/14/2023]
Abstract
We employ a typical genetic circuit model to explore how noise can influence dynamic structure. With the increase of a key interactive parameter, the model will deterministically go through two bifurcations and three dynamic structure regions. We find that a quasiperiodic component, which is not allowed by deterministic dynamics, will be generated by noise inducing in the first two regions, and this quasiperiod will be more and more stable along with the increase in noise. In particular, in the second region the quasiperiod will compete with a stable limit cycle and perform a new transient rhythm. Furthermore, we ascertain the entropy production rate and the heat dissipation rate, and discover a minimal value with theoretical elucidation. In the end, we unveil the mechanism of the formation of quasiperiods, and show a practical biological example. We expect this work to be helpful in solving some biological or ecological problems, such as the genetic origin of periodical cicadas and population dynamics with fluctuation.
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Affiliation(s)
- Yuxuan Wu
- Biophysics & Complex System Center, Center of Theoretical Physics, College of Physics, Jilin University Changchun 130012, People's Republic of China
| | - Yuxing Jiao
- Biophysics & Complex System Center, Center of Theoretical Physics, College of Physics, Jilin University Changchun 130012, People's Republic of China
| | - Yanzhen Zhao
- Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - Haojun Jia
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Liufang Xu
- Biophysics & Complex System Center, Center of Theoretical Physics, College of Physics, Jilin University Changchun 130012, People's Republic of China
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8
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Niemeyer N, Schleimer JH, Schreiber S. Biophysical models of intrinsic homeostasis: Firing rates and beyond. Curr Opin Neurobiol 2021; 70:81-88. [PMID: 34454303 DOI: 10.1016/j.conb.2021.07.011] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2021] [Revised: 06/14/2021] [Accepted: 07/14/2021] [Indexed: 12/01/2022]
Abstract
In view of ever-changing conditions both in the external world and in intrinsic brain states, maintaining the robustness of computations poses a challenge, adequate solutions to which we are only beginning to understand. At the level of cell-intrinsic properties, biophysical models of neurons permit one to identify relevant physiological substrates that can serve as regulators of neuronal excitability and to test how feedback loops can stabilize crucial variables such as long-term calcium levels and firing rates. Mathematical theory has also revealed a rich set of complementary computational properties arising from distinct cellular dynamics and even shaping processing at the network level. Here, we provide an overview over recently explored homeostatic mechanisms derived from biophysical models and hypothesize how multiple dynamical characteristics of cells, including their intrinsic neuronal excitability classes, can be stably controlled.
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Affiliation(s)
- Nelson Niemeyer
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, 10115, Berlin, Germany; Einstein Center for Neurosciences Berlin, Charitéplatz 1, 10117, Berlin, Germany; Bernstein Center for Computational Neuroscience, 10115, Berlin, Germany
| | - Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, 10115, Berlin, Germany; Bernstein Center for Computational Neuroscience, 10115, Berlin, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, 10115, Berlin, Germany; Einstein Center for Neurosciences Berlin, Charitéplatz 1, 10117, Berlin, Germany; Bernstein Center for Computational Neuroscience, 10115, Berlin, Germany.
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9
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Contreras SA, Schleimer JH, Gulledge AT, Schreiber S. Activity-mediated accumulation of potassium induces a switch in firing pattern and neuronal excitability type. PLoS Comput Biol 2021; 17:e1008510. [PMID: 34043638 PMCID: PMC8205125 DOI: 10.1371/journal.pcbi.1008510] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2020] [Revised: 06/15/2021] [Accepted: 04/16/2021] [Indexed: 01/30/2023] Open
Abstract
During normal neuronal activity, ionic concentration gradients across a neuron’s membrane are often assumed to be stable. Prolonged spiking activity, however, can reduce transmembrane gradients and affect voltage dynamics. Based on mathematical modeling, we investigated the impact of neuronal activity on ionic concentrations and, consequently, the dynamics of action potential generation. We find that intense spiking activity on the order of a second suffices to induce changes in ionic reversal potentials and to consistently induce a switch from a regular to an intermittent firing mode. This transition is caused by a qualitative alteration in the system’s voltage dynamics, mathematically corresponding to a co-dimension-two bifurcation from a saddle-node on invariant cycle (SNIC) to a homoclinic orbit bifurcation (HOM). Our electrophysiological recordings in mouse cortical pyramidal neurons confirm the changes in action potential dynamics predicted by the models: (i) activity-dependent increases in intracellular sodium concentration directly reduce action potential amplitudes, an effect typically attributed solely to sodium channel inactivation; (ii) extracellular potassium accumulation switches action potential generation from tonic firing to intermittently interrupted output. Thus, individual neurons may respond very differently to the same input stimuli, depending on their recent patterns of activity and/or the current brain-state. Ionic concentrations in the brain are not constant. We show that during intense neuronal activity, they can change on the order of seconds and even switch neuronal spiking patterns under identical stimulation from a regular firing mode to an intermittently interrupted one. Triggered by an accumulation of extracellular potassium, such a transition is caused by a specific, qualitative change in of the neuronal voltage dynamics—a so-called bifurcation—which affects crucial features of action-potential generation and bears consequences for how information is encoded and how neurons behave together in the network. Also, changes in intracellular sodium can induce measurable effects, like a reduction of spike amplitude that occurs independently of the fast amplitude effects attributed to sodium channel inactivation. Taken together, our results demonstrate that a neuron can respond very differently to the same stimulus, depending on its previous activity or the current brain state. This finding may be particularly relevant when other regulatory mechanisms of ionic homeostasis are challenged, for example, during pathological states of glial impairment or oxygen deprivation. Finally, categorization of cortical neurons as intrinsically bursting or regular spiking may be biased by the ionic concentrations at the time of the observation, highlighting the non-static nature of neuronal dynamics.
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Affiliation(s)
- Susana Andrea Contreras
- Institute for Theoretical Biology, Humboldt-University of Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
| | - Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Humboldt-University of Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
| | - Allan T. Gulledge
- Molecular and Systems Biology, Geisel School of Medicine at Dartmouth College, Hanover, New Hampshire, United States of America
| | - Susanne Schreiber
- Institute for Theoretical Biology, Humboldt-University of Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
- * E-mail:
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10
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Schleimer JH, Hesse J, Contreras SA, Schreiber S. Firing statistics in the bistable regime of neurons with homoclinic spike generation. Phys Rev E 2021; 103:012407. [PMID: 33601551 DOI: 10.1103/physreve.103.012407] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2019] [Accepted: 11/20/2020] [Indexed: 11/07/2022]
Abstract
Neuronal voltage dynamics of regularly firing neurons typically has one stable attractor: either a fixed point (like in the subthreshold regime) or a limit cycle that defines the tonic firing of action potentials (in the suprathreshold regime). In two of the three spike onset bifurcation sequences that are known to give rise to all-or-none type action potentials, however, the resting-state fixed point and limit cycle spiking can coexist in an intermediate regime, resulting in bistable dynamics. Here, noise can induce switches between the attractors, i.e., between rest and spiking, and thus increase the variability of the spike train compared to neurons with only one stable attractor. Qualitative features of the resulting spike statistics depend on the spike onset bifurcations. This paper focuses on the creation of the spiking limit cycle via the saddle-homoclinic orbit (HOM) bifurcation and derives interspike interval (ISI) densities for a conductance-based neuron model in the bistable regime. The ISI densities of bistable homoclinic neurons are found to be unimodal yet distinct from the inverse Gaussian distribution associated with the saddle-node-on-invariant-cycle bifurcation. It is demonstrated that for the HOM bifurcation the transition between rest and spiking is mainly determined along the downstroke of the action potential-a dynamical feature that is not captured by the commonly used reset neuron models. The deduced spike statistics can help to identify HOM dynamics in experimental data.
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Affiliation(s)
- Jan-Hendrik Schleimer
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany
| | - Janina Hesse
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- MSH Medical School Hamburg, Am Kaiserkai 1, 20457 Hamburg, Germany
| | - Susana Andrea Contreras
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany
| | - Susanne Schreiber
- Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Philippstrasse 13, Haus 4, 10115 Berlin, Germany
- Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany
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11
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Marcotte CD, Biktashev VN. Predicting critical ignition in slow-fast excitable models. Phys Rev E 2020; 101:042201. [PMID: 32422766 DOI: 10.1103/physreve.101.042201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2019] [Accepted: 03/06/2020] [Indexed: 11/07/2022]
Abstract
Linearization around unstable traveling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable traveling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable traveling wave solution converges to a stationary "critical nucleus" of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.
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Affiliation(s)
- Christopher D Marcotte
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, EX4 4QJ Exeter, United Kingdom
| | - Vadim N Biktashev
- Department of Mathematics, University of Exeter, EX4 4QF Exeter, United Kingdom
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12
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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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