Paul C. Bressloff

Spatiotemporal dynamics of continuum neural fields

We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and Turing-like patterns.

The Role of Feedback in Shaping the Extra-Classical Receptive Field of Cortical Neurons: A Recurrent Network Model

The responses of neurons in sensory cortices are affected by the spatial context within which stimuli are embedded. In the primary visual cortex (V1), orientation-selective responses to stimuli in the receptive field (RF) center are suppressed by similarly oriented stimuli in the RF surround. Surround suppression, a likely neural correlate of perceptual figure–ground segregation, is traditionally thought to be generated within V1 by long-range horizontal connections. Recently however, it has been shown that these connections are too short and too slow to mediate fast suppression from distant regions of the RF surround. We use an anatomically and physiologically constrained recurrent network model of macaque V1 to show how interareal feedback connections, which are faster and longer-range than horizontal connections, can generate “far” surround suppression. We provide a novel solution to the puzzle of how surround suppression can arise from excitatory feedback axons contacting predominantly excitatory neurons in V1. The basic mechanism involves divergent feedback connections from the far surround targeting pyramidal neurons sending monosynaptic horizontal connections to excitatory and inhibitory neurons in the RF center. One of several predictions of our model is that the “suppressive far surround” is not always suppressive, but can facilitate the response of the RF center, depending on the amount of excitatory drive to the local inhibitors. Our model provides a general mechanism of how top-down feedback signals directly contribute to generating cortical neuron responses to simple sensory stimuli.

What geometric visual hallucinations tell us about the visual cortex

Many observers see geometric visual hallucinations after taking hallucinogens such as LSD, cannabis, mescaline or psilocybin; on viewing bright flickering lights; on waking up or falling asleep; in near-death experiences; and in many other syndromes. Klver organized the images into four groups called form constants: (I) tunnels and funnels, (II) spirals, (III) lattices, including honeycombs and triangles, and (IV) cobwebs. In most cases, the images are seen in both eyes and move with them. We interpret this to mean that they are generated in the brain. Here, we summarize a theory of their origin in visual cortex (area V1), based on the assumption that the form of the retinocortical map and the architecture of V1 determine their geometry. (A much longer and more detailed mathematical version has been published in Philosophical Transactions of the Royal Society B, 356 [2001].) We model V1 as the continuum limit of a lattice of interconnected hypercolumns, each comprising a number of interconnected iso-orientation columns. Based on anatomical evidence, we assume that the lateral connectivity between hypercolumns exhibits symmetries, rendering it invariant under the action of the Euclidean group E(2), composed of reflections and translations in the plane, and a (novel) shift-twist action. Using this symmetry, we show that the various patterns of activity that spontaneously emerge when V1s spatially uniform resting state becomes unstable correspond to the form constants when transformed to the visual field using the retino-cortical map. The results are sensitive to the detailed specification of the lateral connectivity and suggest that the cortical mechanisms that generate geometric visual hallucinations are closely related to those used to process edges, contours, surfaces, and textures.

Traveling fronts and wave propagation failure in an inhomogeneous neural network

We use averaging and homogenization theory to study the propagation of traveling wavefronts in an inhomogeneous excitable neural medium. Motivated by the functional architecture of primary visual cortex, we model the inhomogeneity as a periodic modulation in the long-range neuronal connections. We derive an expression for the effective wavespeed and show that propagation failure can occur if the speed is too slow or the degree of inhomogeneity is too large. We find that there are major qualitative differences in the wavespeed for different choices of the homogenized weight distribution.

Breathing Pulses in an Excitatory Neural Network

In this paper we show how a local inhomogeneous input can stabilize a stationary-pulse solution in an excitatory neural network. A subsequent reduction of the input amplitude can then induce a Hopf instability of the stationary solution resulting in the formation of a breather. The breather can itself undergo a secondary instability leading to the periodic emission of traveling waves. In one dimension such waves consist of pairs of counterpropagating pulses, whereas in two dimensions the waves are circular target patterns.

Dynamics of Strongly Coupled Spiking Neurons

We present a dynamical theory of integrate-and-fire neurons with strong synaptic coupling. We show how phase-locked states that are stable in the weak coupling regime can destabilize as the coupling is increased, leading to states characterized by spatiotemporal variations in the interspike intervals (ISIs). The dynamics is compared with that of a corresponding network of analog neurons in which the outputs of the neurons are taken to be mean firing rates. A fundamental result is that for slow interactions, there is good agreement between the two models (on an appropriately defined timescale). Various examples of desynchronization in the strong coupling regime are presented. First, a globally coupled network of identical neurons with strong inhibitory coupling is shown to exhibit oscillator death in which some of the neurons suppress the activity of others. However, the stability of the synchronous state persists for very large networks and fast synapses. Second, an asymmetric network with a mixture of excitation and inhibition is shown to exhibit periodic bursting patterns. Finally, a one-dimensional network of neurons with long-range interactions is shown to desynchronize to a state with a spatially periodic pattern of mean firing rates across the network. This is modulated by deterministic fluctuations of the instantaneous firing rate whose size is an increasing function of the speed of synaptic response.

Stochastic Neural Field Theory and the System-Size Expansion

We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit (

Low firing-rates in a compartmental model neuron

N\rightarrow\infty

Stochastic Processes in Cell Biology

) we recover standard activity-based or voltage-based rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen system-size expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at

Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

\mathcal{O}(1/N)