Current Topic Workshop: Nonlocal Integro-Differential Equations in Mathematics and Biology

(March 6,2003 - March 8,2003 )

Organizers


Bjorn Sandstede
Department of Mathematics, The Ohio State University
David Terman
Department of Mathematics, The Ohio State University

The goal of the workshop is to articulate the present and future trends in:

  1. Modeling the neuronal networks
  2. Analysis of related mathematical models, and to discuss
  3. General mathematical techniques for integro-differential equations.

The nonlocal interaction of neurons plays a crucial role in the generation of waves and patterns in the brain. Each neuron may send excitatory or inhibitory synaptic input to other neurons, and the intrinsic and synaptic dynamics may involve multiple time scales. The spatio-temporal properties of patterns, such as whether neurons fire in synchrony or not, typically depend on the type and strength of the neuronal interactions.

The mathematical models that incorporate these effects are often integro-differential equations. These models account for the spatial interactions via convolution integrals whose kernels encode the specific interaction properties of neurons. Not much analytical work has been done for equations of this type, and special properties such as maximum principles or singular perturbation theory need to be exploited.

This workshop focuses on surveying the mathematical techniques used to model and analyze the inherently nonlocal interaction of neurons and on concrete applications in neuroscience.

Accepted Speakers

Shun-ichi Amari
Riken Brain Science Institute
Peter Bates
Mathematics Department, Michigan State University
Amit Bose
Courant Institute of Mathematical Sciences, New York University
Carson Chow
Department of Mathematics, University of Pittsburgh
Stephen Coombes
Dept. of Mathematical Sciences, Loughborough University
Bard Ermentrout
Department of Mathematics, University of Pittsburgh
David Golomb
Department of Physiology, Ben-Gurion University of the Negev
Wendy Hines
School of Mathematics, Georgia Institute of Technology
Tim Lewis
Center for Neural Science, New York University
Gabriel Lord
Department of Mathematics, Heriot-Watt University
Jonathan Rubin
Department of Mathematics, University of Pittsburgh
Bjorn Sandstede
Department of Mathematics, The Ohio State University
William Troy
Mathematics Department, University of Pittsburgh
Carl van Vreeswijk
CNRS UMR 8119, University Rene Descartes
Linghai Zhang
Department of Computer Science, York University
Thursday, March 6, 2003
Time Session
09:30 AM
10:30 AM
Shun-ichi Amari - Dynamics of Excitation and Self-Organization in Neural Fields

The present talk summarizes the techniques of boundary layers to understand the dynamical behaviors of neural excitation and self-organization in neural continua. The techniques are rather old, but they can be applied to contemporary problems, including oscillations and synchronization.



  1. Amari, S. (1977). Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics, 27, 77-87.

  2. Kishimoto, K., & Amari, S. (1997). Existence and stability of local excitations in homogeneous neural fields. Journal of Mathematical Biology, 7, 303-318.

  3. Takeuchi, A., & Amari, S. (1979). Formation of topographic maps and columnar microstructures. Biological Cybernetics, 35, 63-72.

  4. Amari, S. (1980). Topographic organization of nerve fields. Bulletin of Mathematical Biology, 42, 339-364.

  5. Amari, S. (1983). Field theory of self-organizing neural nets. IEEE Trans.Systems, Man and Cybernetics, 13(9)(10), 741-748.

11:00 AM
12:00 PM
Bard Ermentrout - Integral Equations in Neurobiology

In this talk, I will describe a number of different integral equations which arise from problems in neuroscience and then give a brief survey of some of the mathematical techniques that have been applied to them. I will start with a simple model for topographic organization and derive a linear integral equation. The mathematical question concerns proof that it has an eigenvalue of 1. The second problem deals with a continuum of coupled oscillators and one wants to show the existence of a phase-locked solution. The third problem deals with the existence of traveling fronts in a neural network. If time permits, I will describe some other methods for dealing with these nonlocal equations.


References



  1. Ermentrout, G.B. (1992). Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM Journal on Applied Mathematics, 52(6), 1665-1687.

  2. Ermentrout, G.B., & McLeod, B. (1993). Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edinburgh Sect. A, 123(3), 461-478.

  3. Curtu, R., & Ermentrout, G.B. (2001). Oscillations in a refractory neural net. Journal of Mathematical Biology, 43, 81-100.

  4. Pinto, D., & Ermentrout, G.B. (2001), Spatially structured activity in synaptically coupled neuronal networks: I. traveling fronts and pulses. SIAM Journal on Applied Mathematics, 62(1), 206-225.

  5. Eveson, S. P. (1991). An integral equation arising from a problem in mathematical biology. Bull. London Math. Soc., 23(3), 293-299.

  6. Golomb, D., & Ermentrout, G.B. (2000). Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity. Network-Comp Neural, 11(3), 221-246.

02:00 PM
02:30 PM
Amit Bose - Synchronization in Globally Inhibitory Networks

In this talk, I will discuss synchronization in a globally inhibitory network that is loosely based on the CA1-CA3 hippocampal regions. Specifically, I will show how transiently increasing the inhibitory frequency greatly expands the basin of attraction of the synchronous solution provided that the inhibition is depressing. The results to be described are local, but this region of the brain also displays fast synchronization over larger distances. Mechanisms that may be responsible for spatial synchronization will be noted, together with possible areas for additional mathematical exploration. This work also suggests that networks with short-term synaptic plasticity exhibit functionally relevant transient phenomena that may be important to consider in other neuronal modeling studies, be they local or non-local.

02:30 PM
03:00 PM
Tim Lewis - Wave Propagation in Networks of Fast-spiking Interneurons

Fast-spiking (FS) interneurons in the cortex are connected by both direct electrical coupling and recurrent inhibitory synapses. The high level of excitability of FS interneurons (in vivo) and the extensive electrical coupling between cells provides appropriate conditions for propagated waves of excitation in the FS cell network. On the other hand, the synaptic inhibition in the network opposes this activity. In my talk, I will formulate an idealized model of FS network that takes the form of an integro-partial differential equation, and I will present preliminary results on wave propagation in the model.

03:30 PM
04:00 PM
David Golomb - Stability of Asynchronized State in Networks of Electrically Coupled Neurons

Electrical coupling may stabilize an asynchronous state in a network of electrically coupled neurons. To determine the conditions that lead to stable asynchronized states, we study a model of quadratic integrate-and-fire neurons and calculate the stability of the asynchronized state for every coupling strength. We obtain an algebraic-integral eigenvalue equation that is solved by converting it to a boundary-value problem.

Friday, March 7, 2003
Time Session
09:00 AM
10:00 AM
Carson Chow - Existence and Stability of Localized Pulses in Neuronal Networks with Nonsaturating Gain

We consider localized pulse solutions in a one-dimensional integro-differential equation similar to that proposed by Amari but with a non-saturating piecewise linear gain function. We show how the equation arises from a network of coupled spiking neurons. The existence condition for pulses can be reduced to the solution of an algebraic system and using this condition we map out the shape of the pulses for differing weight kernels and gain slopes. We also find conditions for the existence of pulses with a 'dimple' on top and for double pulses. A condition for stability of the pulses is also derived.

10:30 AM
11:30 AM
Bjorn Sandstede - Evans Functions for Equations with Nonlocal Terms

The Evans function is a useful tool for the stability analysis of nonlinear waves in partial differential equations on unbounded domains. Its main purpose is to help to locate point spectrum, ie isolated eigenvalues, of the relevant linearized operator. While its computation is typically only possible in perturbative situations or when additional structure such as slow-fast spatial scales is present, the Evans function also provides a parity index that is easier to compute and gives a sufficient condition for instability. In this talk, I will begin with a brief overview of the Evans function and some of its applications. I will then present a recent extension of the Evans-function framework to equations that contain nonlocal terms of a certain form (including those that arise frequently in nonlinear optics and in models of neuronal networks with nonlocal interaction).

01:30 PM
02:00 PM
Stephen Coombes - Travelling Waves in Neural Field Theories with Space-dependent Delays

I plan to talk about the analysis of travelling fronts in neural field theories which incorporate delays arising from the finite speed of action potential propagation. I will discuss i) exact solutions for linear firing rate functions, ii) methods of obtaining numerical solutions for sigmoidal firing rates, and iii) exact solutions for Heaviside firing rate functions.


Aspects of this talk are shortly to appear in a paper:
Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, by S Coombes, G J Lord and M R Owen, Physica D Mar 2003.


You can download a preprint from:
http://www.lboro.ac.uk/departments/ma/preprints/papers02/02-43abs.html

02:00 PM
02:30 PM
William Troy - PDE Methods For Nonlocal Models

We develop PDE methods to study the formation of mutli-bumps in a partial integro differential equation in two space dimensions. We derive a PDE which is equivalent to the integral equation. We then look for circularly symmetric statioanry solutions of the PDE. The linearization of the PDE around these solutions provides a criterion for their stability. When a solution is unstable our analysis predicts the number of peaks that form when the solution of the PDE is a small perturbation from the circulary symmetric solution. We illustrate our results with specific numerical examples.


This work is a joint effort with Carlo Laing.

03:00 PM
03:30 PM
Jonathan Rubin - Localized Activity without Recurrent Excitatory Connections

It is known that the combination of recurrent excitation and lateral inhibition can produce localized "bumps" of sustained activity in integro-differential firing rate models. However, certain populations of reciprocally connected excitatory and inhibitory cells lack connections between excitatory cells. I will discuss the existence of bumps in models without recurrent excitation, highlighting several new features that emerge. I will also consider how spatial variations in coupling affect the existence of bumps.

03:30 PM
04:00 PM
Carl van Vreeswijk - Analysis of the Asynchronous State in a Spatially Extended Network of Spiking Neurons

In the analysis of large spatially extended networks of neurons the neurral field approach is often taken. This approach assumes that all neurons, that are close to each other in space, will fire at approximately the same time. But we know that all-to-all coupled networks (networks in which all neurons are at effectively the same position) can evolve to a state in which different neurons fire at different times. For all-to-all coupled network of identical neurons, in which the coupling strength scales as one over the number of cells, the input I_i into cell i is the same for all cells, I-i=I, and one can write a Fokker-Planck equation for the distribution rho(V) of the voltage, V_i, of the cells. This suggests that in a spatially extended network the input is, in the large N limit, a continuous function of the position, I_i=I(x_i), where x_i is the position of cell i, and one can write a Fokker-Plank equation rho(V,x) for a distribution of voltages that depends continuously on the position x.


I will show that this is indeed the case, and that the resulting Fokker-Planck equation is a partial integro-differential equation. If the system is transltionally invariant, the asynchronous state can be determined, and its stability analysed.

Saturday, March 8, 2003
Time Session
09:00 AM
10:00 AM
Peter Bates - Patterns and Waves for Discrete and Continuum Bistable Equations with Indefinite Interaction

Patterns and Waves for Discrete and Continuum Bistable Equations with Indefinite Interaction

10:30 AM
11:00 AM
Wendy Hines - Convergence to Equilibrium for a Nonlocal Reaction Diffusion Equation

Convergence to Equilibrium for a Nonlocal Reaction Diffusion Equation

11:00 AM
11:30 AM
Gabriel Lord - A Stochastic Baer and Rinzel Model with Spatially Smooth Noise

We consider travelling waves in a stochastically forced Baer and Rinzel model of distributed dendritic spines along a diffusive cable. Formally this is written as a system of stochastic differential equations which should more properly be interpreted as an integral system. The stochastic forcing is taken as white in time. In space we vary the regularity of the noise from being spatially smooth (in fact in a Gevrey class) to white in space. Spatially smooth noise can be interpreted as non-local forcing with some correlation length.


In this talk I plan to introduce the basic model, the stochastic forcing and derive a numerical scheme that preserves the regularity. Numerical results will examine the effect of changing smoothness of the stochastic forcing in space and the noise level in the system.

11:30 AM
12:00 PM
Linghai Zhang - Asymptotic Stability of Traveling Pulse Solutions arising from Neuronal Networks

We study the asymptotic stability of traveling wave solutions of nonlinear systems of integral differential equations. It has been established that nonlinear stability of traveling waves is equivalent to linear stability. Moreover if max for some positive constant is an algebraically simple eigenvalue of L, then the linear stability follows, where L is the operator obtained by linearizing a nonlinear system about its traveling wave and sigma(L) is the spectrum of L. The main aim of this paper is to construct Evans function for determining eigenvalues of operators regarding traveling wave stability.


When considering multipulse solutions, certain components of the traveling waves cross their thresholds for many times. These crossings cause great difficulty in the construction of Evans function. In particular we have to solve an overdetermined system to define the Evans function. By using the method of variation of parameters and by investigating boundedness on of eigenfunction candidates, we find a way to define the Evans function. The zeros of the Evans function coincide with the eigenvalues of the operator L.


By estimating the zeros of the Evans function, we establish the asymptotic stability of the traveling wave of an example from synaptically coupled neuronal networks, describing spatially structured activity.

Name Email Affiliation
Amari, Shun-ichi amari@brain.riken.go.jp Riken Brain Science Institute
Bates, Peter bates@math.msu.edu Mathematics Department, Michigan State University
Borisovich, Andrei Institute of Mathematics, University of Gdansk
Borisyuk, Alla borisyuk@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Bose, Amitabha amitabha.k.bose@njit.edu Courant Institute of Mathematical Sciences, New York University
Chow, Carson carsonc@mail.nih.gov Department of Mathematics, University of Pittsburgh
Coombes, Stephen stephen.coombes@nottingham.ac.uk Dept. of Mathematical Sciences, Loughborough University
Cowen, Carl cowen@mbi.osu.edu Department of Mathematics, The Ohio State University
Cracium, Gheorghe craciun@math.wisc.edu Mathematical Biosciences Institute, The Ohio State University
Danthi, Sanjay danthi.1@osu.edu Mathematical Biosciences Institute, The Ohio State University
Dougherty, Daniel dpdoughe@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Ermentrout, Bard bard@pitt.edu Department of Mathematics, University of Pittsburgh
Fall, Chris fall@uic.edu Center for Neural Science, New York University
French, Donald french@math.uc.edu Dept. of Mathematical Sciences, University of Cincinnati
Golomb, David golomb@bgumail.bgu.ac.il Department of Physiology, Ben-Gurion University of the Negev
Guo, Yixin yixin@math.drexel.edu Department of Mathematics, University of Pittsburgh
Hayot, Fernand hayot@mps.ohio-state.edu Department of Physics, The Ohio State University
Hines, Wendy ghines@math.unl.edu School of Mathematics, Georgia Institute of Technology
Horn, Mary Ann mhorn@nsf.gov Department of Mathematics, Vanderbilt University
Krisner, Edward edkst3@euler.math.pitt.edu University of Pittsburgh
Lewis, Tim tim.lewis@nyu.edu Center for Neural Science, New York University
Lord, Gabriel gabriel@ma.hw.ac.uk Department of Mathematics, Heriot-Watt University
Nykamp, Duane nykamp@math.ucla.edu
Rejniak, Katarzyna rejniak@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Rubin, Jonathan rubin@euler.math.pitt.edu Department of Mathematics, University of Pittsburgh
Sandstede, Bjorn sandsted@math.ohio-state.edu Department of Mathematics, The Ohio State University
Smith, Gregory greg@as.wm.edu Dept. of Applied Science, College of William and Mary
Terman, David terman@math.ohio-state.edu Department of Mathematics, The Ohio State University
Thomson, Mitchell Mathematical Biosciences Institute, The Ohio State University
Trost, Craig craig.trost@pfizer.com Computational Medicine, Pfizer Central Research
Troy, William troy@math.pitt.edu Mathematics Department, University of Pittsburgh
van Vreeswijk, Carl carl.van-vreeswijk@biomedicale.univ-paris5.fr CNRS UMR 8119, University Rene Descartes
Wechselberger, Martin wm@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Yew, Alice yew@math.ohio-state.edu Department of Mathematics, The Ohio State University
Zhang, Linghai liz5@lehigh.edu Department of Computer Science, York University
Dynamics of Excitation and Self-Organization in Neural Fields

The present talk summarizes the techniques of boundary layers to understand the dynamical behaviors of neural excitation and self-organization in neural continua. The techniques are rather old, but they can be applied to contemporary problems, including oscillations and synchronization.



  1. Amari, S. (1977). Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics, 27, 77-87.

  2. Kishimoto, K., & Amari, S. (1997). Existence and stability of local excitations in homogeneous neural fields. Journal of Mathematical Biology, 7, 303-318.

  3. Takeuchi, A., & Amari, S. (1979). Formation of topographic maps and columnar microstructures. Biological Cybernetics, 35, 63-72.

  4. Amari, S. (1980). Topographic organization of nerve fields. Bulletin of Mathematical Biology, 42, 339-364.

  5. Amari, S. (1983). Field theory of self-organizing neural nets. IEEE Trans.Systems, Man and Cybernetics, 13(9)(10), 741-748.

Patterns and Waves for Discrete and Continuum Bistable Equations with Indefinite Interaction

Patterns and Waves for Discrete and Continuum Bistable Equations with Indefinite Interaction

Synchronization in Globally Inhibitory Networks

In this talk, I will discuss synchronization in a globally inhibitory network that is loosely based on the CA1-CA3 hippocampal regions. Specifically, I will show how transiently increasing the inhibitory frequency greatly expands the basin of attraction of the synchronous solution provided that the inhibition is depressing. The results to be described are local, but this region of the brain also displays fast synchronization over larger distances. Mechanisms that may be responsible for spatial synchronization will be noted, together with possible areas for additional mathematical exploration. This work also suggests that networks with short-term synaptic plasticity exhibit functionally relevant transient phenomena that may be important to consider in other neuronal modeling studies, be they local or non-local.

Existence and Stability of Localized Pulses in Neuronal Networks with Nonsaturating Gain

We consider localized pulse solutions in a one-dimensional integro-differential equation similar to that proposed by Amari but with a non-saturating piecewise linear gain function. We show how the equation arises from a network of coupled spiking neurons. The existence condition for pulses can be reduced to the solution of an algebraic system and using this condition we map out the shape of the pulses for differing weight kernels and gain slopes. We also find conditions for the existence of pulses with a 'dimple' on top and for double pulses. A condition for stability of the pulses is also derived.

Travelling Waves in Neural Field Theories with Space-dependent Delays

I plan to talk about the analysis of travelling fronts in neural field theories which incorporate delays arising from the finite speed of action potential propagation. I will discuss i) exact solutions for linear firing rate functions, ii) methods of obtaining numerical solutions for sigmoidal firing rates, and iii) exact solutions for Heaviside firing rate functions.


Aspects of this talk are shortly to appear in a paper:
Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, by S Coombes, G J Lord and M R Owen, Physica D Mar 2003.


You can download a preprint from:
http://www.lboro.ac.uk/departments/ma/preprints/papers02/02-43abs.html

Integral Equations in Neurobiology

In this talk, I will describe a number of different integral equations which arise from problems in neuroscience and then give a brief survey of some of the mathematical techniques that have been applied to them. I will start with a simple model for topographic organization and derive a linear integral equation. The mathematical question concerns proof that it has an eigenvalue of 1. The second problem deals with a continuum of coupled oscillators and one wants to show the existence of a phase-locked solution. The third problem deals with the existence of traveling fronts in a neural network. If time permits, I will describe some other methods for dealing with these nonlocal equations.


References



  1. Ermentrout, G.B. (1992). Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM Journal on Applied Mathematics, 52(6), 1665-1687.

  2. Ermentrout, G.B., & McLeod, B. (1993). Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edinburgh Sect. A, 123(3), 461-478.

  3. Curtu, R., & Ermentrout, G.B. (2001). Oscillations in a refractory neural net. Journal of Mathematical Biology, 43, 81-100.

  4. Pinto, D., & Ermentrout, G.B. (2001), Spatially structured activity in synaptically coupled neuronal networks: I. traveling fronts and pulses. SIAM Journal on Applied Mathematics, 62(1), 206-225.

  5. Eveson, S. P. (1991). An integral equation arising from a problem in mathematical biology. Bull. London Math. Soc., 23(3), 293-299.

  6. Golomb, D., & Ermentrout, G.B. (2000). Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity. Network-Comp Neural, 11(3), 221-246.

Stability of Asynchronized State in Networks of Electrically Coupled Neurons

Electrical coupling may stabilize an asynchronous state in a network of electrically coupled neurons. To determine the conditions that lead to stable asynchronized states, we study a model of quadratic integrate-and-fire neurons and calculate the stability of the asynchronized state for every coupling strength. We obtain an algebraic-integral eigenvalue equation that is solved by converting it to a boundary-value problem.

Convergence to Equilibrium for a Nonlocal Reaction Diffusion Equation

Convergence to Equilibrium for a Nonlocal Reaction Diffusion Equation

Wave Propagation in Networks of Fast-spiking Interneurons

Fast-spiking (FS) interneurons in the cortex are connected by both direct electrical coupling and recurrent inhibitory synapses. The high level of excitability of FS interneurons (in vivo) and the extensive electrical coupling between cells provides appropriate conditions for propagated waves of excitation in the FS cell network. On the other hand, the synaptic inhibition in the network opposes this activity. In my talk, I will formulate an idealized model of FS network that takes the form of an integro-partial differential equation, and I will present preliminary results on wave propagation in the model.

A Stochastic Baer and Rinzel Model with Spatially Smooth Noise

We consider travelling waves in a stochastically forced Baer and Rinzel model of distributed dendritic spines along a diffusive cable. Formally this is written as a system of stochastic differential equations which should more properly be interpreted as an integral system. The stochastic forcing is taken as white in time. In space we vary the regularity of the noise from being spatially smooth (in fact in a Gevrey class) to white in space. Spatially smooth noise can be interpreted as non-local forcing with some correlation length.


In this talk I plan to introduce the basic model, the stochastic forcing and derive a numerical scheme that preserves the regularity. Numerical results will examine the effect of changing smoothness of the stochastic forcing in space and the noise level in the system.

Localized Activity without Recurrent Excitatory Connections

It is known that the combination of recurrent excitation and lateral inhibition can produce localized "bumps" of sustained activity in integro-differential firing rate models. However, certain populations of reciprocally connected excitatory and inhibitory cells lack connections between excitatory cells. I will discuss the existence of bumps in models without recurrent excitation, highlighting several new features that emerge. I will also consider how spatial variations in coupling affect the existence of bumps.

Evans Functions for Equations with Nonlocal Terms

The Evans function is a useful tool for the stability analysis of nonlinear waves in partial differential equations on unbounded domains. Its main purpose is to help to locate point spectrum, ie isolated eigenvalues, of the relevant linearized operator. While its computation is typically only possible in perturbative situations or when additional structure such as slow-fast spatial scales is present, the Evans function also provides a parity index that is easier to compute and gives a sufficient condition for instability. In this talk, I will begin with a brief overview of the Evans function and some of its applications. I will then present a recent extension of the Evans-function framework to equations that contain nonlocal terms of a certain form (including those that arise frequently in nonlinear optics and in models of neuronal networks with nonlocal interaction).

PDE Methods For Nonlocal Models

We develop PDE methods to study the formation of mutli-bumps in a partial integro differential equation in two space dimensions. We derive a PDE which is equivalent to the integral equation. We then look for circularly symmetric statioanry solutions of the PDE. The linearization of the PDE around these solutions provides a criterion for their stability. When a solution is unstable our analysis predicts the number of peaks that form when the solution of the PDE is a small perturbation from the circulary symmetric solution. We illustrate our results with specific numerical examples.


This work is a joint effort with Carlo Laing.

Analysis of the Asynchronous State in a Spatially Extended Network of Spiking Neurons

In the analysis of large spatially extended networks of neurons the neurral field approach is often taken. This approach assumes that all neurons, that are close to each other in space, will fire at approximately the same time. But we know that all-to-all coupled networks (networks in which all neurons are at effectively the same position) can evolve to a state in which different neurons fire at different times. For all-to-all coupled network of identical neurons, in which the coupling strength scales as one over the number of cells, the input I_i into cell i is the same for all cells, I-i=I, and one can write a Fokker-Planck equation for the distribution rho(V) of the voltage, V_i, of the cells. This suggests that in a spatially extended network the input is, in the large N limit, a continuous function of the position, I_i=I(x_i), where x_i is the position of cell i, and one can write a Fokker-Plank equation rho(V,x) for a distribution of voltages that depends continuously on the position x.


I will show that this is indeed the case, and that the resulting Fokker-Planck equation is a partial integro-differential equation. If the system is transltionally invariant, the asynchronous state can be determined, and its stability analysed.

Asymptotic Stability of Traveling Pulse Solutions arising from Neuronal Networks

We study the asymptotic stability of traveling wave solutions of nonlinear systems of integral differential equations. It has been established that nonlinear stability of traveling waves is equivalent to linear stability. Moreover if max for some positive constant is an algebraically simple eigenvalue of L, then the linear stability follows, where L is the operator obtained by linearizing a nonlinear system about its traveling wave and sigma(L) is the spectrum of L. The main aim of this paper is to construct Evans function for determining eigenvalues of operators regarding traveling wave stability.


When considering multipulse solutions, certain components of the traveling waves cross their thresholds for many times. These crossings cause great difficulty in the construction of Evans function. In particular we have to solve an overdetermined system to define the Evans function. By using the method of variation of parameters and by investigating boundedness on of eigenfunction candidates, we find a way to define the Evans function. The zeros of the Evans function coincide with the eigenvalues of the operator L.


By estimating the zeros of the Evans function, we establish the asymptotic stability of the traveling wave of an example from synaptically coupled neuronal networks, describing spatially structured activity.