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Biophysics of ComputationInformation Processing in Single Neurons$
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Christof Koch

Print publication date: 1998

Print ISBN-13: 9780195104912

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195104912.001.0001

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The Membrane Equation

The Membrane Equation

Chapter:
(p.5) 1 The Membrane Equation
Source:
Biophysics of Computation
Author(s):

Christof Koch

Publisher:
Oxford University Press
DOI:10.1093/oso/9780195139853.003.0007

Any physical or biophysical mechanism instantiating an information processing system that needs to survive in the real world must obey several constraints: (1) it must operate at high speeds, (2) it must have a rich repertoire of computational primitives, with the ability to implement a variety of linear and nonlinear, high-gain, operations, and (3) it must interface with the physical world—in the sense of being able to represent sensory input patterns accurately and translate the result of the computations into action, that is motor output (Keyes, 1985). The membrane potential is the one physical variable within the nervous system that fulfills these three requirements: it can vary rapidly over large distances (e.g., an action potential changes the potential by 100 mV within 1 msec, propagating up to 1 cm or more down an axon within that time), and the membrane potential controls a vast number of nonlinear gates—ionic channels—that provide a very rich substrate for implementing nonlinear operations. These channels transduce visual, tactile, auditory, and olfactory stimuli into thanges of the membrane potential, and such voltage changes back into the release of neurotransmitters or the contraction of muscles. This is not to deny that ionic fluxes, or chemical interactions of various substances with each other, are not crucial to the working of the brain. They are, and we will study some of these mechanisms in Chap. 11. Yet the membrane potential is the incisive variable that serves as primary vehicle for the neuronal operations underlying rapid computations—at the fraction of a second time scale—in the brain. We will introduce the reader in a very gentle manner to the electrical properties of nerve cells by starting off with the very simplest of all neuronal models, consisting of nothing more than a resistance and a capacitance (a so-called RC circuit). Yet endowed with synaptic input, this model can already implement a critical nonlinear operation, divisive normalization and gain control. As a starting point, we choose a so-called point representation of a neuron. Here, the spatial dependency of the neuron is reduced to a single point or compartment.

Keywords:   Action potentials, Capacitance, Depolarization, Extracellular potential, Gain control, Hyperpolarization, Input admittance, Leak conductance, Membrane capacitance, Neuromuscular junction

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