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.
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.
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.
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:
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.
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.
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.
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.
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.
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).
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.
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.
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.