Organizers
The goal of the workshop is to articulate the present and future trends in:
 Modeling the neuronal networks
 Analysis of related mathematical models, and to discuss
 General mathematical techniques for integrodifferential 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 spatiotemporal 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 integrodifferential 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
Thursday, March 6, 2003  

Time  Session 
09:30 AM 10:30 AM  Shunichi Amari  Dynamics of Excitation and SelfOrganization in Neural Fields The present talk summarizes the techniques of boundary layers to understand the dynamical behaviors of neural excitation and selforganization in neural continua. The techniques are rather old, but they can be applied to contemporary problems, including oscillations and synchronization.

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 phaselocked 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

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 CA1CA3 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 shortterm synaptic plasticity exhibit functionally relevant transient phenomena that may be important to consider in other neuronal modeling studies, be they local or nonlocal. 
02:30 PM 03:00 PM  Tim Lewis  Wave Propagation in Networks of Fastspiking Interneurons Fastspiking (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 integropartial 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 integrateandfire neurons and calculate the stability of the asynchronized state for every coupling strength. We obtain an algebraicintegral eigenvalue equation that is solved by converting it to a boundaryvalue 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 onedimensional integrodifferential equation similar to that proposed by Amari but with a nonsaturating 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 slowfast 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 Evansfunction 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 Spacedependent 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: You can download a preprint from: 
02:00 PM 02:30 PM  William Troy  PDE Methods For Nonlocal Models We develop PDE methods to study the formation of mutlibumps 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 integrodifferential 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 alltoall 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 alltoall 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, Ii=I, and one can write a FokkerPlanck 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 FokkerPlank 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 FokkerPlanck equation is a partial integrodifferential 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 nonlocal 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  Affiliation  

Amari, Shunichi  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, BenGurion University of the Negev 
Guo, Yixin  yixin@math.drexel.edu  Department of Mathematics, University of Pittsburgh 
Hayot, Fernand  hayot@mps.ohiostate.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, HeriotWatt 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.ohiostate.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.ohiostate.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.vanvreeswijk@biomedicale.univparis5.fr  CNRS UMR 8119, University Rene Descartes 
Wechselberger, Martin  wm@mbi.osu.edu  Mathematical Biosciences Institute, The Ohio State University 
Yew, Alice  yew@math.ohiostate.edu  Department of Mathematics, The Ohio State University 
Zhang, Linghai  liz5@lehigh.edu  Department of Computer Science, York University 
The present talk summarizes the techniques of boundary layers to understand the dynamical behaviors of neural excitation and selforganization in neural continua. The techniques are rather old, but they can be applied to contemporary problems, including oscillations and synchronization.
 Amari, S. (1977). Dynamics of pattern formation in lateralinhibition type neural fields. Biological Cybernetics, 27, 7787.
 Kishimoto, K., & Amari, S. (1997). Existence and stability of local excitations in homogeneous neural fields. Journal of Mathematical Biology, 7, 303318.
 Takeuchi, A., & Amari, S. (1979). Formation of topographic maps and columnar microstructures. Biological Cybernetics, 35, 6372.
 Amari, S. (1980). Topographic organization of nerve fields. Bulletin of Mathematical Biology, 42, 339364.
 Amari, S. (1983). Field theory of selforganizing neural nets. IEEE Trans.Systems, Man and Cybernetics, 13(9)(10), 741748.
Patterns and Waves for Discrete and Continuum Bistable Equations with Indefinite Interaction
In this talk, I will discuss synchronization in a globally inhibitory network that is loosely based on the CA1CA3 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 shortterm synaptic plasticity exhibit functionally relevant transient phenomena that may be important to consider in other neuronal modeling studies, be they local or nonlocal.
We consider localized pulse solutions in a onedimensional integrodifferential equation similar to that proposed by Amari but with a nonsaturating 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.
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 axodendritic 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/papers02/0243abs.html
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 phaselocked 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
 Ermentrout, G.B. (1992). Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM Journal on Applied Mathematics, 52(6), 16651687.
 Ermentrout, G.B., & McLeod, B. (1993). Existence and uniqueness of travelling waves for a neural network. Proc. Roy. Soc. Edinburgh Sect. A, 123(3), 461478.
 Curtu, R., & Ermentrout, G.B. (2001). Oscillations in a refractory neural net. Journal of Mathematical Biology, 43, 81100.
 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), 206225.
 Eveson, S. P. (1991). An integral equation arising from a problem in mathematical biology. Bull. London Math. Soc., 23(3), 293299.
 Golomb, D., & Ermentrout, G.B. (2000). Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity. NetworkComp Neural, 11(3), 221246.
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 integrateandfire neurons and calculate the stability of the asynchronized state for every coupling strength. We obtain an algebraicintegral eigenvalue equation that is solved by converting it to a boundaryvalue problem.
Convergence to Equilibrium for a Nonlocal Reaction Diffusion Equation
Fastspiking (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 integropartial differential equation, and I will present preliminary results on wave propagation in the model.
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 nonlocal 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.
It is known that the combination of recurrent excitation and lateral inhibition can produce localized "bumps" of sustained activity in integrodifferential 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.
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 slowfast 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 Evansfunction 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).
We develop PDE methods to study the formation of mutlibumps 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.
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 alltoall 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 alltoall 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, Ii=I, and one can write a FokkerPlanck 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 FokkerPlank 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 FokkerPlanck equation is a partial integrodifferential equation. If the system is transltionally invariant, the asynchronous state can be determined, and its stability analysed.
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.