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Current Topics Workshop Abstracts and Lecture Materials:
Author: Shun-ichi
Amari, RIKEN Brain Science Institute Presentation Materials: PPT 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.
Author: Peter
Bates, Michigan State University
Author: Amitabha
Bose, Courant Institute of Mathematical Sciences 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. Author: Carson
Chow , University of Pittsburgh Presentation
Materials: PDF 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.
Author: Steve
Coombes , University of Pittsburgh Presentation Materials: PDF 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:
Author: Bard
Ermentrout , University of Pittsburgh Presentation Materials: PDF 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. Author: David
Golomb , Ben-Gurion University Presentation Materials: PPT 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.
Author: Gwendolen
Hines, Georgia Inst. of Technology
Author: Tim
Lewis , Center for Neural Science 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. Author: Gabriel
Lord, Heriot-Watt University Presentation Materials: PDF 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.
Author: Jonathan
Rubin, University of Pittsburgh Presentation Materials: PDF 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.
Author: Bjorn
Sandstede, The Ohio State University Presentation Materials: PDF1
PDF2 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).
Author: William
Troy , University of Pittsburgh Presentation Materials: PDF 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)
Author: Linghai
Zhang, Lehigh University 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.
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