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Abstract
Controlling the excess and shortage of energy is a fundamental task for living organisms. Diabetes is a representative metabolic disease caused by the malfunction of energy homeostasis. The islets of Langerhans in the pancreas release long-range messengers, hormones, into the blood to regulate the homeostasis of the primary energy fuel, glucose. The hormone and glucose levels in the blood show rhythmic oscillations with a characteristic period of 5-10 min, and the functional roles of the oscillations are not clear. Each islet has [Formula: see text] and [Formula: see text] cells that secrete glucagon and insulin, respectively. These two counter-regulatory hormones appear sufficient to increase and decrease glucose levels. However, pancreatic islets have a third cell type, [Formula: see text] cells, which secrete somatostatin. The three cell populations have a unique spatial organization in islets, and they interact to perturb their hormone secretions. The mini-organs of islets are scattered throughout the exocrine pancreas. Considering that the human pancreas contains approximately a million islets, the coordination of hormone secretion from the multiple sources of islets and cells within the islets should have a significant effect on human physiology. In this review, we introduce the hierarchical organization of tripartite cell networks, and recent biophysical modeling to systematically understand the oscillations and interactions of [Formula: see text], [Formula: see text], and [Formula: see text] cells. Furthermore, we discuss the functional roles and clinical implications of hormonal oscillations and their phase coordination for the diagnosis of type II diabetes.
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Affiliation(s)
- Taegeun Song
- Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk 37673, Republic of Korea
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Lee B, Song T, Lee K, Kim J, Han S, Berggren PO, Ryu SH, Jo J. Phase modulation of insulin pulses enhances glucose regulation and enables inter-islet synchronization. PLoS One 2017; 12:e0172901. [PMID: 28235104 PMCID: PMC5325581 DOI: 10.1371/journal.pone.0172901] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2016] [Accepted: 02/10/2017] [Indexed: 11/23/2022] Open
Abstract
Insulin is secreted in a pulsatile manner from multiple micro-organs called the islets of Langerhans. The amplitude and phase (shape) of insulin secretion are modulated by numerous factors including glucose. The role of phase modulation in glucose homeostasis is not well understood compared to the obvious contribution of amplitude modulation. In the present study, we measured Ca2+ oscillations in islets as a proxy for insulin pulses, and we observed their frequency and shape changes under constant/alternating glucose stimuli. Here we asked how the phase modulation of insulin pulses contributes to glucose regulation. To directly answer this question, we developed a phenomenological oscillator model that drastically simplifies insulin secretion, but precisely incorporates the observed phase modulation of insulin pulses in response to glucose stimuli. Then, we mathematically modeled how insulin pulses regulate the glucose concentration in the body. The model of insulin oscillation and glucose regulation describes the glucose-insulin feedback loop. The data-based model demonstrates that the existence of phase modulation narrows the range within which the glucose concentration is maintained through the suppression/enhancement of insulin secretion in conjunction with the amplitude modulation of this secretion. The phase modulation is the response of islets to glucose perturbations. When multiple islets are exposed to the same glucose stimuli, they can be entrained to generate synchronous insulin pulses. Thus, we conclude that the phase modulation of insulin pulses is essential for glucose regulation and inter-islet synchronization.
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Affiliation(s)
- Boah Lee
- Department of Life Sciences, Pohang University of Science and Technology, Pohang, Gyeongbuk, Korea
| | - Taegeun Song
- Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk, Korea
| | - Kayoung Lee
- School of Interdisciplinary Bioscience and Bioengineering, Pohang University of Science and Technology, Pohang, Gyeongbuk, Korea
| | - Jaeyoon Kim
- The Rolf Luft Research Center for Diabetes and Endocrinology, Karolinska Institute, Stockholm, Sweden
| | - Seungmin Han
- School of Interdisciplinary Bioscience and Bioengineering, Pohang University of Science and Technology, Pohang, Gyeongbuk, Korea
| | - Per-Olof Berggren
- School of Interdisciplinary Bioscience and Bioengineering, Pohang University of Science and Technology, Pohang, Gyeongbuk, Korea
- The Rolf Luft Research Center for Diabetes and Endocrinology, Karolinska Institute, Stockholm, Sweden
| | - Sung Ho Ryu
- Department of Life Sciences, Pohang University of Science and Technology, Pohang, Gyeongbuk, Korea
- School of Interdisciplinary Bioscience and Bioengineering, Pohang University of Science and Technology, Pohang, Gyeongbuk, Korea
| | - Junghyo Jo
- Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk, Korea
- Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk, Korea
- * E-mail:
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Sun Y, Zhao D. Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks. CHAOS (WOODBURY, N.Y.) 2012; 22:023131. [PMID: 22757538 DOI: 10.1063/1.4721997] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We study the effect of noise on the outer synchronization between two unidirectionally coupled complex networks and find analytically that outer synchronization could be achieved via white-noise-based coupling. It is also demonstrated that, if two networks have both conventional linear coupling and white-noise-based coupling, the critical deterministic coupling strength between two complex networks for synchronization transition decreases with an increase in the intensity of noise. We provide numerical results to illustrate the feasibility and effectiveness of the theoretical results.
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Affiliation(s)
- Yongzheng Sun
- School of Sciences, China University of Mining and Technology, Xuzhou 221008, People's Republic of China.
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4
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History-dependent excitability as a single-cell substrate of transient memory for information discrimination. PLoS One 2010; 5:e15023. [PMID: 21203387 PMCID: PMC3010997 DOI: 10.1371/journal.pone.0015023] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2010] [Accepted: 10/08/2010] [Indexed: 11/19/2022] Open
Abstract
Neurons react differently to incoming stimuli depending upon their previous history of stimulation. This property can be considered as a single-cell substrate for transient memory, or context-dependent information processing: depending upon the current context that the neuron "sees" through the subset of the network impinging on it in the immediate past, the same synaptic event can evoke a postsynaptic spike or just a subthreshold depolarization. We propose a formal definition of History-Dependent Excitability (HDE) as a measure of the propensity to firing in any moment in time, linking the subthreshold history-dependent dynamics with spike generation. This definition allows the quantitative assessment of the intrinsic memory for different single-neuron dynamics and input statistics. We illustrate the concept of HDE by considering two general dynamical mechanisms: the passive behavior of an Integrate and Fire (IF) neuron, and the inductive behavior of a Generalized Integrate and Fire (GIF) neuron with subthreshold damped oscillations. This framework allows us to characterize the sensitivity of different model neurons to the detailed temporal structure of incoming stimuli. While a neuron with intrinsic oscillations discriminates equally well between input trains with the same or different frequency, a passive neuron discriminates better between inputs with different frequencies. This suggests that passive neurons are better suited to rate-based computation, while neurons with subthreshold oscillations are advantageous in a temporal coding scheme. We also address the influence of intrinsic properties in single-cell processing as a function of input statistics, and show that intrinsic oscillations enhance discrimination sensitivity at high input rates. Finally, we discuss how the recognition of these cell-specific discrimination properties might further our understanding of neuronal network computations and their relationships to the distribution and functional connectivity of different neuronal types.
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Nakao H, Teramae JN, Goldobin DS, Kuramoto Y. Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise. CHAOS (WOODBURY, N.Y.) 2010; 20:033126. [PMID: 20887066 DOI: 10.1063/1.3488977] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
An effective white-noise Langevin equation is derived that describes long-time phase dynamics of a limit-cycle oscillator driven by weak stationary colored noise. Effective drift and diffusion coefficients are given in terms of the phase sensitivity of the oscillator and the correlation function of the noise, and are explicitly calculated for oscillators with sinusoidal phase sensitivity functions driven by two typical colored Gaussian processes. The results are verified by numerical simulations using several types of stochastic or chaotic noise. The drift and diffusion coefficients of oscillators driven by chaotic noise exhibit anomalous dependence on the oscillator frequency, reflecting the peculiar power spectrum of the chaotic noise.
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Affiliation(s)
- Hiroya Nakao
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
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Arai K, Nakao H. Averaging approach to phase coherence of uncoupled limit-cycle oscillators receiving common random impulses. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:066220. [PMID: 19256938 DOI: 10.1103/physreve.78.066220] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2008] [Indexed: 05/27/2023]
Abstract
Populations of uncoupled limit-cycle oscillators receiving common random impulses show various types of phase-coherent states, which are characterized by the distribution of phase differences between pairs of oscillators. We develop a theory to predict the stationary distribution of pairwise phase differences from the phase response curve, which quantitatively encapsulates the oscillator dynamics, via averaging of the Frobenius-Perron equation describing the impulse-driven oscillators. The validity of our theory is confirmed by direct numerical simulations using the FitzHugh-Nagumo neural oscillator receiving common Poisson impulses as an example.
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Affiliation(s)
- Kensuke Arai
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan.
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7
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Arai K, Nakao H. Phase coherence in an ensemble of uncoupled limit-cycle oscillators receiving common Poisson impulses. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:036218. [PMID: 18517496 DOI: 10.1103/physreve.77.036218] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2007] [Revised: 12/09/2007] [Indexed: 05/26/2023]
Abstract
An ensemble of uncoupled limit-cycle oscillators receiving common Poisson impulses shows a range of nontrivial behavior, from synchronization, desynchronization, to clustering. The group behavior that arises in the ensemble can be predicted from the phase response of a single oscillator to a given impulsive perturbation. We present a theory based on phase reduction of a jump stochastic process describing a Poisson-driven limit-cycle oscillator, and verify the results through numerical simulations and electric circuit experiments. We also give a geometrical interpretation of the synchronizing mechanism, a perturbative expansion to the stationary phase distribution, and the diffusion limit of our jump stochastic model.
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Affiliation(s)
- Kensuke Arai
- Department of Physics, Kyoto University, Kyoto, Japan.
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Nakao H, Arai K, Kawamura Y. Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators. PHYSICAL REVIEW LETTERS 2007; 98:184101. [PMID: 17501578 DOI: 10.1103/physrevlett.98.184101] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2006] [Indexed: 05/15/2023]
Abstract
We study synchronization properties of general uncoupled limit-cycle oscillators driven by common and independent Gaussian white noises. Using phase reduction and averaging methods, we analytically derive the stationary distribution of the phase difference between oscillators for weak noise intensity. We demonstrate that in addition to synchronization, clustering, or more generally coherence, always results from arbitrary initial conditions, irrespective of the details of the oscillators.
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Affiliation(s)
- Hiroya Nakao
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
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Tateno T, Robinson HPC. Phase resetting curves and oscillatory stability in interneurons of rat somatosensory cortex. Biophys J 2007; 92:683-95. [PMID: 17192317 PMCID: PMC1751383 DOI: 10.1529/biophysj.106.088021] [Citation(s) in RCA: 112] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Synchronous oscillations in neural activity are found over wide areas of the cortex. Specific populations of interneurons are believed to play a significant role in generating these synchronized oscillations through mutual synaptic and gap-junctional interactions. Little is known, though, about the mechanism of how oscillations are maintained stably by particular types of interneurons and by their local networks. To obtain more insight into this, we measured membrane-potential responses to small current-pulse perturbations during regular firing, to construct phase resetting curves (PRCs) for three types of interneurons: nonpyramidal regular-spiking (NPRS), low-threshold spiking (LTS), and fast-spiking (FS) cells. Within each cell type, both monophasic and biphasic PRCs were observed, but the proportions and sensitivities to perturbation amplitude were clearly correlated to cell type. We then analyzed the experimentally measured PRCs to predict oscillation stability, or firing reliability, of cells for a complex stochastic input, as occurs in vivo. To do this, we used a method from random dynamical system theory to estimate Lyapunov exponents of the simplified phase model on the circle. The results indicated that LTS and NPRS cells have greater oscillatory stability (are more reliably entrained) in small noisy inputs than FS cells, which is consistent with their distinct types of threshold dynamics.
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Affiliation(s)
- T Tateno
- Department of Physiology, University of Cambridge, Cambridge, United Kingdom
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Tateno T, Robinson HPC. Quantifying noise-induced stability of a cortical fast-spiking cell model with Kv3-channel-like current. Biosystems 2006; 89:110-6. [PMID: 17433533 DOI: 10.1016/j.biosystems.2006.05.016] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2005] [Accepted: 05/22/2006] [Indexed: 11/17/2022]
Abstract
Population oscillations in neural activity in the gamma (>30 Hz) and higher frequency ranges are found over wide areas of the mammalian cortex. Recently, in the somatosensory cortex, the details of neural connections formed by several types of GABAergic interneurons have become apparent, and they are believed to play a significant role in generating these oscillations through synaptic and gap-junctional interactions. However, little is known about the mechanism of how such oscillations are maintained stably by particular interneurons and by their local networks, in a noisy environment with abundant synaptic inputs. To obtain more insight into this, we studied a fast-spiking (FS)-cell model including Kv3-channel-like current, which is a distinctive feature of these cells, from the viewpoint of nonlinear dynamical systems. To examine the specific role of the Kv3-channel in determining oscillation properties, we analyzed basic properties of the FS-cell model, such as the bifurcation structure and phase resetting curves (PRCs). Furthermore, to quantitatively characterize the oscillation stability under noisy fluctuations mimicking small fast synaptic inputs, we applied a recently developed method from random dynamical system theory to estimate Lyapunov exponents, both for the original four-dimensional dynamics and for a reduced one-dimensional phase-equation on the circle. The results indicated that the presence of the Kv3-channel-like current helps to regulate the stability of noisy neural oscillations and a transient-period length to stochastic attractors.
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Affiliation(s)
- T Tateno
- Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, Osaka, Japan.
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Nakao H, Arai KS, Nagai K, Tsubo Y, Kuramoto Y. Synchrony of limit-cycle oscillators induced by random external impulses. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:026220. [PMID: 16196697 DOI: 10.1103/physreve.72.026220] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2005] [Indexed: 05/04/2023]
Abstract
The mechanism of phase synchronization between uncoupled limit-cycle oscillators induced by common random impulsive forcing is analyzed. By reducing the dynamics of the oscillator to a random phase map, it is shown that phase synchronization generally occurs when the oscillator is driven by weak random impulsive forcing in the limit of large interimpulse intervals. The case where the interimpulse intervals are finite is also analyzed perturbatively for small impulse intensity. For weak Poisson impulses, it is shown that the phase synchronization persists up to the first order approximation.
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Affiliation(s)
- Hiroya Nakao
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan.
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Nagai K, Nakao H, Tsubo Y. Synchrony of neural oscillators induced by random telegraphic currents. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:036217. [PMID: 15903556 DOI: 10.1103/physreve.71.036217] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2004] [Indexed: 05/02/2023]
Abstract
When a neuron receives a randomly fluctuating input current, its reliability of spike generation improves compared with the case of a constant input current [Mainen and Sejnowski, Science 268, 1503 (1995)]. This phenomenon can be interpreted as phase synchronization between uncoupled nonlinear oscillators subject to a common external input. We analyze this phenomenon using dynamical models of neurons, assuming the input current to be a simple random telegraphic signal that jumps between two values, and the neuron to be always purely self-oscillatory. The internal state of the neuron randomly jumps between two limit cycles corresponding to the input values, which can be described by random phase maps when the switching time of the input current is sufficiently long. Using such a random map description, we discuss the synchrony of neural oscillators subject to fluctuating inputs. Especially when the phase maps are monotonic, we can generally show that the Lyapunov exponent is negative, namely, phase synchronization is stable and reproducibility of spike timing improves.
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Affiliation(s)
- Ken Nagai
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan
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13
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Tateno T, Pakdaman K. Random dynamics of the Morris-Lecar neural model. CHAOS (WOODBURY, N.Y.) 2004; 14:511-530. [PMID: 15446961 DOI: 10.1063/1.1756118] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Determining the response characteristics of neurons to fluctuating noise-like inputs similar to realistic stimuli is essential for understanding neuronal coding. This study addresses this issue by providing a random dynamical system analysis of the Morris-Lecar neural model driven by a white Gaussian noise current. Depending on parameter selections, the deterministic Morris-Lecar model can be considered as a canonical prototype for widely encountered classes of neuronal membranes, referred to as class I and class II membranes. In both the transitions from excitable to oscillating regimes are associated with different bifurcation scenarios. This work examines how random perturbations affect these two bifurcation scenarios. It is first numerically shown that the Morris-Lecar model driven by white Gaussian noise current tends to have a unique stationary distribution in the phase space. Numerical evaluations also reveal quantitative and qualitative changes in this distribution in the vicinity of the bifurcations of the deterministic system. However, these changes notwithstanding, our numerical simulations show that the Lyapunov exponents of the system remain negative in these parameter regions, indicating that no dynamical stochastic bifurcations take place. Moreover, our numerical simulations confirm that, regardless of the asymptotic dynamics of the deterministic system, the random Morris-Lecar model stabilizes at a unique stationary stochastic process. In terms of random dynamical system theory, our analysis shows that additive noise destroys the above-mentioned bifurcation sequences that characterize class I and class II regimes in the Morris-Lecar model. The interpretation of this result in terms of neuronal coding is that, despite the differences in the deterministic dynamics of class I and class II membranes, their responses to noise-like stimuli present a reliable feature.
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Affiliation(s)
- Takashi Tateno
- Department of Physiology, University of Cambridge, Downing Street, Cambridge CB2 3EG, United Kingdom.
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Ritt J. Evaluation of entrainment of a nonlinear neural oscillator to white noise. ACTA ACUST UNITED AC 2003; 68:041915. [PMID: 14682981 DOI: 10.1103/physreve.68.041915] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/23/2003] [Indexed: 11/07/2022]
Abstract
The Lyapunov exponent for a one-dimensional neural oscillator model, the theta neuron, is computed for white noise forcing, using the steady-state solution to the associated Fokker-Planck equation. The latter is mildly singular, due to the nature of the multiplicative input. In agreement with previous results with similar models, the exponent is negative for all forcing amplitudes, but here it is shown to be small, relative to that for periodic drive, in a range of forcing strengths. Thus the synchronization of an ensemble of independent neurons receiving common but random input can be slow. Moreover, this implies that aperiodic input may be suboptimal, in some contexts, for preserving the reliability of fine spike timing, a potentially important component of the neural "code."
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Affiliation(s)
- Jason Ritt
- McGovern Institute for Brain Research, MIT E25-414, Cambridge, Massachusetts 02139, USA.
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Hunter JD, Milton JG. Amplitude and frequency dependence of spike timing: implications for dynamic regulation. J Neurophysiol 2003; 90:387-94. [PMID: 12634276 DOI: 10.1152/jn.00074.2003] [Citation(s) in RCA: 56] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022] Open
Abstract
The spike-time reliability of motoneurons in the Aplysia buccal motor ganglion was studied as a function of the frequency content and the relative amplitude of the fluctuations in the neuronal input, calculated as the coefficient of variation (CV). Measurements of spike-time reliability to sinusoidal and aperiodic inputs, as well as simulations of a noisy leaky integrate-and-fire neuron stimulated by spike trains drawn from a periodically modulated process, demonstrate that there are three qualitatively different CV-dependent mechanisms that determine reliability: noise-dominated (CV < 0.05 for Aplysia motoneurons) where spike timing is unreliable regardless of frequency content; resonance-dominated (CV approximately 0.05-0.25) where reliability is reduced by removal of input frequencies equal to motoneuron firing rate; and amplitude-dominated (CV >0.35) where reliability depends on input frequencies greater than motoneuron firing rate. In the resonance-dominated regime, changes in the activity of the presynaptic inhibitory interneuron B4/5 alter motoneuron spike-time reliability. The increases or decreases in reliability occur coincident with small changes in motoneuron spiking rate due to changes in interneuron activity. Injection of a hyperpolarizing current into the motoneuron reproduces the interneuron-induced changes in reliability. The rate-dependent changes in reliability can be understood from the phase-locking properties of regularly spiking motoneurons to periodic inputs. Our observations demonstrate that the ability of a neuron to support a spike-time code can be actively controlled by varying the properties of the neuron and its input.
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Affiliation(s)
- John D Hunter
- Department of Neurology, University of Chicago, Chicago, Illinois 60615, USA
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Kosmidis EK, Pakdaman K. An analysis of the reliability phenomenon in the FitzHugh-Nagumo model. J Comput Neurosci 2003; 14:5-22. [PMID: 12435921 DOI: 10.1023/a:1021100816798] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes-the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.
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