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Bressloff PC. Truncated stochastically switching processes. Phys Rev E 2024; 109:024103. [PMID: 38491685 DOI: 10.1103/physreve.109.024103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2023] [Accepted: 01/09/2024] [Indexed: 03/18/2024]
Abstract
There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
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2
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Buckwar E, Conte M, Meddah A. A stochastic hierarchical model for low grade glioma evolution. J Math Biol 2023; 86:89. [PMID: 37147527 PMCID: PMC10163130 DOI: 10.1007/s00285-023-01909-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2022] [Revised: 03/17/2023] [Accepted: 03/22/2023] [Indexed: 05/07/2023]
Abstract
A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker-Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.
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Affiliation(s)
- Evelyn Buckwar
- Institute of Stochastics, Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria
- Centre for Mathematical Sciences, Lund University, 221 00, Lund, Sweden
| | - Martina Conte
- Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy
| | - Amira Meddah
- Institute of Stochastics, Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria.
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3
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Bertazzi A, Bierkens J, Dobson P. Approximations of Piecewise Deterministic Markov Processes and their convergence properties. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.09.004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/14/2022]
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4
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Jacak JE, Jacak WA. The Role of Myelin in Malfunctions of Neuron Transmittance. Neuroscience 2022; 505:125-156. [DOI: 10.1016/j.neuroscience.2022.10.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2022] [Revised: 08/29/2022] [Accepted: 10/03/2022] [Indexed: 11/06/2022]
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5
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Pu S, Thomas PJ. Resolving molecular contributions of ion channel noise to interspike interval variability through stochastic shielding. BIOLOGICAL CYBERNETICS 2021; 115:267-302. [PMID: 34021802 DOI: 10.1007/s00422-021-00877-7] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2020] [Accepted: 05/04/2021] [Indexed: 06/12/2023]
Abstract
Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM J Appl Math 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in Phys Rev Lett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."
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Affiliation(s)
- Shusen Pu
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH, USA.
- Department of Biomedical Engineering, Vanderbilt University, Nashville, TN, USA.
| | - Peter J Thomas
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH, USA
- Department of Biology, Case Western Reserve University, Cleveland, OH, USA
- Department of Cognitive Science, Case Western Reserve University, Cleveland, OH, USA
- Department of Data and Computer Science, Case Western Reserve University, Cleveland, OH, USA
- Department of Electrical, Control, and Systems Engineering, Case Western Reserve University, Cleveland, OH, USA
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6
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Barnes CL. Hydrodynamic limit and propagation of chaos for Brownian particles reflecting from a Newtonian barrier. ANN APPL PROBAB 2020. [DOI: 10.1214/19-aap1536] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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7
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Jacak JE, Jacak WA. New wave-type mechanism of saltatory conduction in myelinated axons and micro-saltatory conduction in C fibres. EUROPEAN BIOPHYSICS JOURNAL: EBJ 2020; 49:343-360. [PMID: 32588093 PMCID: PMC7351862 DOI: 10.1007/s00249-020-01442-z] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/19/2020] [Revised: 04/23/2020] [Accepted: 06/02/2020] [Indexed: 01/19/2023]
Abstract
We present a new wave-type model of saltatory conduction in myelinated axons. Poor conductivity in the neuron cytosol limits electrical current signal velocity according to cable theory, to 1–3 m/s, whereas saltatory conduction occurs with a velocity of 100–300 m/s. We propose a wave-type mechanism for saltatory conduction in the form of the kinetics of an ionic plasmon-polariton being the hybrid of the electro-magnetic wave and of the synchronized ionic plasma oscillations in myelinated segments along an axon. The model agrees with observations and allows for description of the regulatory role of myelin. It explains also the mechanism of conduction deficiency in demyelination syndromes such as multiple sclerosis. The recently observed micro-saltatory conduction in ultrathin unmyelinated C fibers with periodic ion gate clusters is also explained.
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Affiliation(s)
- J E Jacak
- Department of Quantum Technology, Wrocław University of Science and Technology, 50-370, Wrocław, Poland.
| | - W A Jacak
- Department of Quantum Technology, Wrocław University of Science and Technology, 50-370, Wrocław, Poland
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Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris–Lecar model. ADV APPL PROBAB 2020. [DOI: 10.1017/apr.2019.55] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
AbstractIn the first part of this paper we study approximations of trajectories of piecewise deterministic processes (PDPs) when the flow is not given explicitly by the thinning method. We also establish a strong error estimate for PDPs as well as a weak error expansion for piecewise deterministic Markov processes (PDMPs). These estimates are the building blocks of the multilevel Monte Carlo (MLMC) method, which we study in the second part. The coupling required by the MLMC is based on the thinning procedure. In the third part we apply these results to a two-dimensional Morris–Lecar model with stochastic ion channels. In the range of our simulations the MLMC estimator outperforms classical Monte Carlo.
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Schmutz V, Gerstner W, Schwalger T. Mesoscopic population equations for spiking neural networks with synaptic short-term plasticity. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:5. [PMID: 32253526 PMCID: PMC7136387 DOI: 10.1186/s13408-020-00082-z] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/11/2019] [Accepted: 03/25/2020] [Indexed: 06/07/2023]
Abstract
Coarse-graining microscopic models of biological neural networks to obtain mesoscopic models of neural activities is an essential step towards multi-scale models of the brain. Here, we extend a recent theory for mesoscopic population dynamics with static synapses to the case of dynamic synapses exhibiting short-term plasticity (STP). The extended theory offers an approximate mean-field dynamics for the synaptic input currents arising from populations of spiking neurons and synapses undergoing Tsodyks-Markram STP. The approximate mean-field dynamics accounts for both finite number of synapses and correlation between the two synaptic variables of the model (utilization and available resources) and its numerical implementation is simple. Comparisons with Monte Carlo simulations of the microscopic model show that in both feedforward and recurrent networks, the mesoscopic mean-field model accurately reproduces the first- and second-order statistics of the total synaptic input into a postsynaptic neuron and accounts for stochastic switches between Up and Down states and for population spikes. The extended mesoscopic population theory of spiking neural networks with STP may be useful for a systematic reduction of detailed biophysical models of cortical microcircuits to numerically efficient and mathematically tractable mean-field models.
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Affiliation(s)
- Valentin Schmutz
- Brain Mind Institute, École Polytechnique Féderale de Lausanne (EPFL), Lausanne, Switzerland.
| | - Wulfram Gerstner
- Brain Mind Institute, École Polytechnique Féderale de Lausanne (EPFL), Lausanne, Switzerland
| | - Tilo Schwalger
- Brain Mind Institute, École Polytechnique Féderale de Lausanne (EPFL), Lausanne, Switzerland
- Bernstein Center for Computational Neuroscience, Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
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10
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Khater MM, Attia RA, Abdel-Aty AH, Abdel-Khalek S, Al-Hadeethi Y, Lu D. On the computational and numerical solutions of the transmission of nerve impulses of an excitable system (the neuron system). JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2020. [DOI: 10.3233/jifs-179547] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Affiliation(s)
| | - Raghda A.M. Attia
- Department of Mathematics, Faculty of Science, Jiangsu University, China
- Department of Basic Science, Higher Technological Institute 10th of Ramadan city, Egypt
| | - Abdel-Haleem Abdel-Aty
- Department of Physics, College of Sciences, University of Bisha, Bisha, Saudi Arabia
- Physics Department, Faculty of Science, Al-Azhar University, Assiut, Egypt
| | - Sayed Abdel-Khalek
- Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
| | - Yas Al-Hadeethi
- Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
| | - Dianchen Lu
- Department of Mathematics, Faculty of Science, Jiangsu University, China
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11
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Fermín LJ, Lévy-Véhel J. Variability and singularity arising from a Piecewise-Deterministic Markov Process applied to model poor patient compliance in the multi-IV case. J Appl Stat 2020; 47:2525-2545. [DOI: 10.1080/02664763.2019.1711030] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Affiliation(s)
- Lisandro J. Fermín
- CIMFAV, Instituto de Ingeniería Matemática, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile
| | - Jacques Lévy-Véhel
- Institut National de Recherche en Informatique et en Automatique (INRIA) Le Chesnay, Le Chesnay, France
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12
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Löcherbach E. Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity. J THEOR PROBAB 2019. [DOI: 10.1007/s10959-019-00947-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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13
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Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes. MATHEMATICS 2019. [DOI: 10.3390/math7060518] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework.
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14
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Grosskinsky S, Jatuviriyapornchai W. Derivation of mean-field equations for stochastic particle systems. Stoch Process Their Appl 2019. [DOI: 10.1016/j.spa.2018.05.006] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
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15
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Bossy M, Fontbona J, Olivero H. Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels. J Math Biol 2019; 78:1771-1820. [PMID: 30734076 DOI: 10.1007/s00285-019-01326-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2018] [Revised: 12/31/2018] [Indexed: 11/24/2022]
Abstract
In this work we are interested in a mathematical model of the collective behavior of a fully connected network of finitely many neurons, when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin-Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean field type. When the leak conductance is strictly positive, we prove that if the initial voltages are uniformly bounded and the electrical interaction between neurons is strong enough, then, uniformly in the number of neurons, the whole system synchronizes exponentially fast as time goes to infinity, up to some error controlled by (and vanishing with) the channels noise level. Moreover, we prove that if the random initial condition is exchangeable, on every bounded time interval the propagation of chaos property for this system holds (regardless of the interaction intensities). Combining these results, we deduce that the nonlinear McKean-Vlasov equation describing an infinite network of such neurons concentrates, as time goes to infinity, around the dynamics of a single Hodgkin-Huxley neuron with chemical neurotransmitter channels. Our results are illustrated and complemented with numerical simulations.
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Affiliation(s)
- Mireille Bossy
- INRIA Sophia Antipolis Méditerranée, Sophia Antipolis, France
| | - Joaquín Fontbona
- Department of Mathematical Engineering and Center for Mathematical Modeling, UMI(2807) UCHILE-CNRS, University of Chile, Santiago, Chile
| | - Héctor Olivero
- CIMFAV, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile.
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16
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Bressloff PC, Maclaurin JN. Stochastic Hybrid Systems in Cellular Neuroscience. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2018; 8:12. [PMID: 30136005 PMCID: PMC6104574 DOI: 10.1186/s13408-018-0067-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/09/2018] [Accepted: 08/05/2018] [Indexed: 06/08/2023]
Abstract
We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space. The latter typically represents some random switching process. We begin by summarizing the basic theory of stochastic hybrid systems, including various approximation schemes in the fast switching (weak noise) limit. In subsequent sections, we consider various applications of stochastic hybrid systems, including stochastic ion channels and membrane voltage fluctuations, stochastic gap junctions and diffusion in randomly switching environments, and intracellular transport in axons and dendrites. Finally, we describe recent work on phase reduction methods for stochastic hybrid limit cycle oscillators.
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17
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León JR, Samson A. Hypoelliptic stochastic FitzHugh–Nagumo neuronal model: Mixing, up-crossing and estimation of the spike rate. ANN APPL PROBAB 2018. [DOI: 10.1214/17-aap1355] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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18
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Bressloff PC, MacLaurin J. A variational method for analyzing limit cycle oscillations in stochastic hybrid systems. CHAOS (WOODBURY, N.Y.) 2018; 28:063105. [PMID: 29960393 DOI: 10.1063/1.5027077] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Many systems in biology can be modeled through ordinary differential equations, which are piece-wise continuous, and switch between different states according to a Markov jump process known as a stochastic hybrid system or piecewise deterministic Markov process (PDMP). In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we develop a phase reduction method for stochastic hybrid systems that support a stable limit cycle in the deterministic limit. A classic example is the Morris-Lecar model of a neuron, where the switching Markov process is the number of open ion channels and the continuous process is the membrane voltage. We outline a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the switching rate ϵ-1. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (-Ca/ϵ).
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
| | - James MacLaurin
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
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19
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Höpfner R, Löcherbach E, Thieullen M. Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems. BERNOULLI 2017. [DOI: 10.3150/16-bej820] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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20
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Hodara P, Krell N, Löcherbach E. Non-parametric estimation of the spiking rate in systems of interacting neurons. STATISTICAL INFERENCE FOR STOCHASTIC PROCESSES 2016. [DOI: 10.1007/s11203-016-9150-4] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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21
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Benaïm M, Le Borgne S, Malrieu F, Zitt PA. Qualitative properties of certain piecewise deterministic Markov processes. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES 2015. [DOI: 10.1214/14-aihp619] [Citation(s) in RCA: 52] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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22
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Riedler MG, Thieullen M. Spatio-temporal hybrid (PDMP) models: Central limit theorem and Langevin approximation for global fluctuations. Application to electrophysiology. BERNOULLI 2015. [DOI: 10.3150/13-bej583] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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23
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Thomas PJ, Lindner B. Asymptotic phase for stochastic oscillators. PHYSICAL REVIEW LETTERS 2014; 113:254101. [PMID: 25554883 DOI: 10.1103/physrevlett.113.254101] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2014] [Indexed: 05/25/2023]
Abstract
Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arise from a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of the asymptotic phase function. In the presence of noise, the asymptotic phase is no longer well defined. We introduce a new definition of asymptotic phase in terms of the slowest decaying modes of the Kolmogorov backward operator. Our stochastic asymptotic phase is well defined for noisy oscillators, even when the oscillations are noise dependent. It reduces to the classical asymptotic phase in the limit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or by empirical observation of an oscillating density's approach to its steady state.
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Affiliation(s)
- Peter J Thomas
- Bernstein Center for Computational Neuroscience, Humboldt University, 10115 Berlin, Germany and Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, Ohio 44106, USA
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience and Department of Physics, Humboldt University, 10115 Berlin, Germany
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24
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Stochastic representations of ion channel kinetics and exact stochastic simulation of neuronal dynamics. J Comput Neurosci 2014; 38:67-82. [PMID: 25408289 DOI: 10.1007/s10827-014-0528-2] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2014] [Revised: 08/18/2014] [Accepted: 09/03/2014] [Indexed: 10/24/2022]
Abstract
In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation with a commonly used numerical approximation strategy. The first representation we present is a random time change representation, popularized by Thomas Kurtz, with the second being analogous to a "Gillespie" representation. Exact stochastic algorithms are provided for the different representations, which are preferable to either (a) fixed time step or (b) piecewise constant propensity algorithms, which still appear in the literature. As examples, we provide versions of the exact algorithms for the Morris-Lecar conductance based model, and detail the error induced, both in a weak and a strong sense, by the use of approximate algorithms on this model. We include ready-to-use implementations of the random time change algorithm in both XPP and Matlab. Finally, through the consideration of parametric sensitivity analysis, we show how the representations presented here are useful in the development of further computational methods. The general representations and simulation strategies provided here are known in other parts of the sciences, but less so in the present setting.
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25
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Cáceres MJ, Perthame B. Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity. J Theor Biol 2014; 350:81-9. [DOI: 10.1016/j.jtbi.2014.02.005] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2013] [Revised: 01/28/2014] [Accepted: 02/06/2014] [Indexed: 10/25/2022]
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26
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Schmerl BA, McDonnell MD. Channel-noise-induced stochastic facilitation in an auditory brainstem neuron model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052722. [PMID: 24329311 DOI: 10.1103/physreve.88.052722] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2013] [Revised: 10/14/2013] [Indexed: 06/03/2023]
Abstract
Neuronal membrane potentials fluctuate stochastically due to conductance changes caused by random transitions between the open and closed states of ion channels. Although it has previously been shown that channel noise can nontrivially affect neuronal dynamics, it is unknown whether ion-channel noise is strong enough to act as a noise source for hypothesized noise-enhanced information processing in real neuronal systems, i.e., "stochastic facilitation". Here we demonstrate that biophysical models of channel noise can give rise to two kinds of recently discovered stochastic facilitation effects in a Hodgkin-Huxley-like model of auditory brainstem neurons. The first, known as slope-based stochastic resonance (SBSR), enables phasic neurons to emit action potentials that can encode the slope of inputs that vary slowly relative to key time constants in the model. The second, known as inverse stochastic resonance (ISR), occurs in tonically firing neurons when small levels of noise inhibit tonic firing and replace it with burstlike dynamics. Consistent with previous work, we conclude that channel noise can provide significant variability in firing dynamics, even for large numbers of channels. Moreover, our results show that possible associated computational benefits may occur due to channel noise in neurons of the auditory brainstem. This holds whether the firing dynamics in the model are phasic (SBSR can occur due to channel noise) or tonic (ISR can occur due to channel noise).
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Affiliation(s)
- Brett A Schmerl
- Computational and Theoretical Neuroscience Laboratory, Institute for Telecommunications Research, University of South Australia, South Australia 5095, Australia
| | - Mark D McDonnell
- Computational and Theoretical Neuroscience Laboratory, Institute for Telecommunications Research, University of South Australia, South Australia 5095, Australia
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Kohno T, Aihara K. Improving noise resistance of intrinsic rhythms in a square-wave burster model. Biosystems 2013; 112:276-83. [PMID: 23541604 DOI: 10.1016/j.biosystems.2013.03.016] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2012] [Revised: 03/01/2013] [Accepted: 03/19/2013] [Indexed: 11/29/2022]
Abstract
The square-wave burster (Wang and Rinzel, 2003) is a class of autonomous bursting cells that share a bifurcation structure. It is known that this class of cells is involved in the generation of various life-supporting rhythms. In our research to realize an electronic circuit that mimics the rhythm generating mechanism in the square-wave burster, our circuit experimentally exhibited severe fluctuations in its rhythmic activity. We have found a noise-sensitive region in the phase portrait of the ideal model and have proposed modifications of the model that can reduce this fluctuation. A possible modification to ionic-conductance neuron models (Kohno and Aihara, 2011) was inspired by them. This modification, however, cannot be applied to a group of square-wave bursters, including the Butera-Rinzel-Smith model (Butera et al., 1999; Del Negro et al., 2001), which is a model of the pre-Bötzinger complex bursting neuron that plays a crucial role in the generation of respiration rhythms, because this modification premises that the slow dynamics originates from an activation gate variable of a hyperpolarizing ionic current. However, in some square-wave bursters, they are controlled by an inactivation gate variable of a depolarizing ionic current. In this study, we proposed a similar modification with a completely different mechanism that can be applied to this group of square-wave bursters. In the presence of noises, the modified Butera-Rinzel-Smith model can generate rhythmic activity that is more stable and similar to biological observations than the original model. The mechanisms underlying this modification are explained with noisy bifurcation diagrams.
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Affiliation(s)
- Takashi Kohno
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan.
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28
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Adiabatic reduction of a model of stochastic gene expression with jump Markov process. J Math Biol 2013; 68:1051-70. [PMID: 23460478 DOI: 10.1007/s00285-013-0661-y] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2012] [Revised: 02/20/2013] [Indexed: 10/27/2022]
Abstract
This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription considered as a jump Markov process. In this model, the process of gene expression with auto-regulation is described by fast/slow dynamics. The production of mRNA is assumed to follow a compound Poisson process occurring at a rate depending on protein levels (the phenomena called bursting in molecular biology) and the production of protein is a linear function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast process (due to faster mRNA degradation than that of protein) we prove that, with appropriate scalings in the burst rate, jump size or translational rate, the bursting phenomena can be transmitted to the slow variable. We show that, depending on the scaling, the reduced equation is either a stochastic differential equation with a jump Poisson process or a deterministic ordinary differential equation. These results are significant because adiabatic reduction techniques seem to have not been rigorously justified for a stochastic differential system containing a jump Markov process. We expect that the results can be generalized to adiabatic methods in more general stochastic hybrid systems.
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29
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Perthame B, Salort D. On a voltage-conductance kinetic system for integrate & fire neural networks. ACTA ACUST UNITED AC 2013. [DOI: 10.3934/krm.2013.6.841] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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30
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Pakdaman K, Thieullen M, Wainrib G. Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic Markov processes. Stoch Process Their Appl 2012. [DOI: 10.1016/j.spa.2012.03.005] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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31
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Baladron J, Fasoli D, Faugeras O, Touboul J. Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2012; 2:10. [PMID: 22657695 PMCID: PMC3497713 DOI: 10.1186/2190-8567-2-10] [Citation(s) in RCA: 51] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2011] [Accepted: 03/09/2012] [Indexed: 05/20/2023]
Abstract
We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the FitzHugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons' initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes place, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is a solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations or non-local partial differential equations resembling the McKean-Vlasov-Fokker-Planck equations. We prove the well-posedness of the McKean-Vlasov equations, i.e. the existence and uniqueness of a solution. We also show the results of some numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiments also indicate that the McKean-Vlasov-Fokker-Planck equations may be a good way to understand the mean-field dynamics through, e.g. a bifurcation analysis.Mathematics Subject Classification (2000): 60F99, 60B10, 92B20, 82C32, 82C80, 35Q80.
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Affiliation(s)
- Javier Baladron
- NeuroMathComp Laboratory, INRIA, Sophia-Antipolis Méditerranée, 06902, France.
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32
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Dangerfield CE, Kay D, Burrage K. Modeling ion channel dynamics through reflected stochastic differential equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051907. [PMID: 23004788 DOI: 10.1103/physreve.85.051907] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2011] [Revised: 02/27/2012] [Indexed: 06/01/2023]
Abstract
Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks.
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Affiliation(s)
- Ciara E Dangerfield
- Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom.
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33
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Riedler M, Thieullen M, Wainrib G. Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models. ELECTRON J PROBAB 2012. [DOI: 10.1214/ejp.v17-1946] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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34
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Benaïm M, Le Borgne S, Malrieu F, Zitt PA. Quantitative ergodicity for some switched dynamical systems. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2012. [DOI: 10.1214/ecp.v17-1932] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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35
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Reduction of stochastic conductance-based neuron models with time-scales separation. J Comput Neurosci 2011; 32:327-46. [DOI: 10.1007/s10827-011-0355-7] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2011] [Revised: 06/13/2011] [Accepted: 07/19/2011] [Indexed: 10/17/2022]
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36
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Bressloff PC, Lai YM. Stochastic synchronization of neuronal populations with intrinsic and extrinsic noise. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2011; 1:2. [PMID: 22656265 PMCID: PMC3280892 DOI: 10.1186/2190-8567-1-2] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2010] [Accepted: 05/03/2011] [Indexed: 05/29/2023]
Abstract
We extend the theory of noise-induced phase synchronization to the case of a neural master equation describing the stochastic dynamics of an ensemble of uncoupled neuronal population oscillators with intrinsic and extrinsic noise. The master equation formulation of stochastic neurodynamics represents the state of each population by the number of currently active neurons, and the state transitions are chosen so that deterministic Wilson-Cowan rate equations are recovered in the mean-field limit. We apply phase reduction and averaging methods to a corresponding Langevin approximation of the master equation in order to determine how intrinsic noise disrupts synchronization of the population oscillators driven by a common extrinsic noise source. We illustrate our analysis by considering one of the simplest networks known to generate limit cycle oscillations at the population level, namely, a pair of mutually coupled excitatory (E) and inhibitory (I) subpopulations. We show how the combination of intrinsic independent noise and extrinsic common noise can lead to clustering of the population oscillators due to the multiplicative nature of both noise sources under the Langevin approximation. Finally, we show how a similar analysis can be carried out for another simple population model that exhibits limit cycle oscillations in the deterministic limit, namely, a recurrent excitatory network with synaptic depression; inclusion of synaptic depression into the neural master equation now generates a stochastic hybrid system.
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Affiliation(s)
- Paul C Bressloff
- Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, UK
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
| | - Yi Ming Lai
- Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, UK
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37
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Buckwar E, Riedler MG. An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J Math Biol 2011; 63:1051-93. [PMID: 21243359 DOI: 10.1007/s00285-010-0395-z] [Citation(s) in RCA: 59] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2010] [Revised: 12/17/2010] [Indexed: 11/28/2022]
Abstract
In this paper, we present a mathematical description for excitable biological membranes, in particular neuronal membranes. We aim to model the (spatio-) temporal dynamics, e.g., the travelling of an action potential along the axon, subject to noise, such as ion channel noise. Using the framework of Piecewise Deterministic Processes (PDPs) we provide an exact mathematical description-in contrast to pseudo-exact algorithms considered in the literature-of the stochastic process one obtains coupling a continuous time Markov chain model with a deterministic dynamic model of a macroscopic variable, that is coupling Markovian channel dynamics to the time-evolution of the transmembrane potential. We extend the existing framework of PDPs in finite dimensional state space to include infinite-dimensional evolution equations and thus obtain a stochastic hybrid model suitable for modelling spatio-temporal dynamics. We derive analytic results for the infinite-dimensional process, such as existence, the strong Markov property and its extended generator. Further, we exemplify modelling of spatially extended excitable membranes with PDPs by a stochastic hybrid version of the Hodgkin-Huxley model of the squid giant axon. Finally, we discuss the advantages of the PDP formulation in view of analytical and numerical investigations as well as the application of PDPs to structurally more complex models of excitable membranes.
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Affiliation(s)
- Evelyn Buckwar
- Department of Mathematics, Maxwell Institute, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom.
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38
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Bressloff PC. Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:051903. [PMID: 21230496 DOI: 10.1103/physreve.82.051903] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/13/2010] [Revised: 09/22/2010] [Indexed: 05/08/2023]
Abstract
We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit N→∞ , where N determines the size of each population, the dynamics is described by deterministic Wilson-Cowan equations. On the other hand, for finite N the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steady-state probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite N , we calculate the exponentially small rate of noise-induced transitions between the resulting metastable states using a Wentzel-Kramers-Brillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory or inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles).
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Affiliation(s)
- Paul C Bressloff
- Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, United Kingdom
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39
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Pakdaman K, Thieullen M, Wainrib G. Diffusion approximation of birth–death processes: Comparison in terms of large deviations and exit points. Stat Probab Lett 2010. [DOI: 10.1016/j.spl.2010.03.006] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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