X-Message-Number: 3915
From: Ralph Merkle <>
Subject: The rate of change of membrane potentials
Date: Mon, 27 Feb 1995 10:56:06 PST

To quote from "Principles of Neural Science," Third edition, by
Kandel, Schwartz and Jessell, page 97:

  The capacitance of the membrane has the effect of reducing the rate
  at which the membrane potential changes in response to a current
  pulse (Figure 7-2a) [7-2a shows a gradual rise of voltage in respone
  to a current pulse].  If the membrane had only resistive properties,
  a step pulse of outward current passed across it would change the
  membrane potential instantaneously (Figure 7-3, line a) [which shows
  a square wave step].  On the other hand, if the membrane had only
  capacitive properties, the membrane potential would change slowly,
  in a ramp-like manner, in response to the same step pulse of current
  (Figure 7-3, line b) [which shows a straight line of moderate slope].
  Because the membrane has *both* capacitive and resistive properties
  in parallel, the actual change in membrane potential resulting from
  a rectangular current pulse combines features of the two pure
  responses.  Thus the initial slope of Vm as a function of time is
  the same as that for a purely capacitive element, whereas the final
  slope and amplitude are the same as those for a purely resistive
  element (Figure 7-3, line c) [which shows a curve gradually approaching
  a limiting value].

  The rising phase of the potential change shown in Figure 7-2A can be
  described by the following equation:

     delta Vm(t) = Im R(1 - e**(-t/tau))

  [Vm is the membrane voltage, Im is the membrane current, t is time].

  where e, which has the value of 2.72, is the base of the system of
  natural logarithms, and tau equals RC, the product of the resistance
  and capacitance of the membrane.  The parameter tau, called the
  *membrane time constant*, can be measured experimentally.  For the
  response of the membrane to a rectangular step of current (Figure 7-3),
  tau is the time that it takes Vm to move 63% of the way toward its
  final value (1-1/e x 100).  The time constants of different neurons
  typically range from 1 to 20 ms.

  The effect of the time constant on integration of synaptic input is
  especially important.  Most synaptic potentials are caused by brief
  synaptic currents triggered by the opening of ligand-gated channels.
  The time course of the rising phase of a synaptic potential is
  determined by both active and passive properties of the membrane,
  but the falling phase is purely a passive process.  Its time course
  is a function of the membrane time constant.  The longer the time
  constant, the longer the duration of the synaptic potential.  When
  synaptic potentials overlap in time, they add together in a process
  known as *temporal summation.*  In this way individual excitatory
  postsynaptic potentials that alone might be too small to trigger an
  action potential can sum to reach threshold.  If a postsynaptic cell
  has a long membrane time constant, the synaptic potential lasts longer
  and there is more chance for temporal summation (Figure 7-4).  [which
  shows various pre-synaptic spike trains, and the variation in the
  time course of the post synaptic membrane voltages as tau is varied].

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