Mathematical neuroscience: from neurons to circuits to systems

doi:10.1016/j.jphysparis.2003.09.005

Journal of Physiology-Paris

Volume 97, Issues 2–3, March–May 2003, Pages 209-219

https://doi.org/10.1016/j.jphysparis.2003.09.005 Get rights and content

Abstract

Applications of mathematics and computational techniques to our understanding of neuronal systems are provided. Reduction of membrane models to simplified canonical models demonstrates how neuronal spike-time statistics follow from simple properties of neurons. Averaging over space allows one to derive a simple model for the whisker barrel circuit and use this to explain and suggest several experiments. Spatio-temporal pattern formation methods are applied to explain the patterns seen in the early stages of drug-induced visual hallucinations.

Introduction

For many years, mathematics and computational methods have played an important role in our understanding of the nervous system. The goal of this chapter is to present some examples of how mathematical techniques can be applied at a variety of levels to increase our understanding of neural systems. We begin with a description of the biophysical principles underlying the generation of action potential in single neurons. The crucial modeling idea is to represent the electrical properties of biological membranes using an equivalent circuit consisting of capacitors and resistors in parallel. We then use phaseplane methods to study a simplified single neuron model and show how the dynamics on the plane can be further reduced to a scalar dynamical system on a circle. Simulations of the reduced model are used to explain the spiking statistics of single neurons driven by noisy stimuli. We then turn our attention to simple neuronal circuits involving networks of excitatory and inhibitory neurons. A mean field approximation reduces such networks to a planar system, and phaseplane analysis enables us to explain experimental results from the somatosensory (touch) system of the rat. Finally, we examine large spatially organized networks. We apply bifurcation theory to these networks and use the results to explain the patterns seen during visual hallucinations.

Section snippets

Equivalent circuit models

The equivalent circuit model has become standard for representing the dynamics of electrical activity observed in single neurons. It is based on the idea that neuronal activity can be completely described by the flow of different currents associated with the neuron’s membrane. Currents are divided into those that can be represented by linear circuit elements (passive currents) and those that are voltage and/or time dependent and require more complex dynamics (active currents). Both sets can be

Phaseplanes and spiking

The Hodgkin-Huxley current-balance model described above is a four-dimensional dynamical system. This makes its rigorous mathematical analysis difficult. Morris and Lecar (see [26]) devised a very simple model neuron based on only three conductances, a fast calcium channel, a slow potassium channel, and a passive leak channel. Using the equivalent circuit formulation described above, the equations are: $C d V d t =I+g_{L} (V_{L} −V)+g_{Ca} m_{∞} (V)(V_{Ca} −V)+g_{K} w(V_{K} −V),$ $d w d t = w_{∞} (V)−w τ_{w} (V) .$

The gating kinetics of the

Statistics of cortical neural activity in vivo

A long standing debate in neuroscience concerns whether neurons code information about the world in terms of the average frequency of spike generation, or through the precise timing of individual spikes. The second hypothesis has been criticized as unlikely since the firing of neurons in the living brain is very irregular (e.g. see [28]). On the other hand, several experiments have shown that neurons are capable of producing reliably timed action potentials. In one particular experiment, it was

Activity models

The previous sections showed how to reduce the current-balance equations to a simple scalar model to understand the dynamics of single neurons. In many experimentally relevant cases, it is desirable to simplify the dynamics still further by using so-called activity or firing rate models. In these models the relevant quantity is not the spike time or potential of an individual neuron, but rather the generalized activity level or firing rate in single neurons or neuronal populations.

In contrast

From cave paintings to the structure of the cortex

For years, people have been fascinated by Paleolithic rock paintings such as are found in the famous Lascaux caves in France and on the sandstone walls in the American Southwest. These pictures often depict animals and human-like images. However, there are also more abstract designs such as spirals, sunbursts and mandalas which occur across all cultures in this art. One of the questions that anthropologists have asked is what are the possible meanings of these abstract symbols. A number of

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Mathematical neuroscience: from neurons to circuits to systems

Abstract

Introduction

Section snippets

Equivalent circuit models

Phaseplanes and spiking

Statistics of cortical neural activity in vivo

Activity models

From cave paintings to the structure of the cortex

Prog. Biophys. Mol. Biol.

Brain Res.

Vis. Res.

Brain Res.

Geometric visual hallucinations, Euclidean symmetry, and the functional architecture of striate cortex

Philos. T Roy. Soc. B

A mathematical theory of visual hallucinations

Kybernetic

Neural Nets as spatio-temporal pattern forming systems

Rep. Prog. Phys.

Spike generating dynamics and the conditions for spike time precision in cortical neurons

J. Comput. Neurosci.

Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics

Neural Comput.