Browse all publications by topic
Browse all publications by year
- J. Kanev, G. Wenning, and
K. Obermayer. Ito Calculus Approach to the Distribution of ISI and
Response-Stimulus Correlation.
.
In Proceedings of the 29th Göttingen Neurobiology Conference,
page 1025, 2003.
Major neural learning mechanisms like spike timing dependant
plasticity, synaptic redistribution and synaptic scaling depend on - and
change - the neurons inter-spike intervall distribution, as well as the
distribution of time differences between stimulus and the neurons response.
The first is identical to the first passage time (FPT) of the membrane
potential, whereas the latter can be expressed as a function of the neurons
mean firing rate and the response-stimulus correlation (in experimental
contexts also called the reverse correlation). In order to gain a deeper
understanding into how these major learning mechanisms work we have a closer
look at the underlying processes which shape the above two distributions. We
try to come up with an analytical expression for these important distribution
densities - the latter of which has up to now only been computed numerically
- aiming for a trade-off between biological realismand mathematical
tractability. We use a leaky-integrate-and-fire neuron with reversal
potentials, in which synaptic inputs are collectively modeled as an
Ornstein-Uhlenbeck process driving the neuron. Neglecting the synaptic time
constants the membrane behaviour satisfies a linear stochastic differential
equation. Using It? Calculus and the method of the integrating factor this
SDE can be solved, and the moments of the solution can be calculated. These
moments are used to approximate the probability of threshold crossings and
thus give an approximate, but analytic solution to the first passage time
problem (FTP), the distribution of inter-spike intervalls (ISI) given the
input. There is no known exact solution to this problem, even for the simple
leaky integrate-and-fire neuron with additive white noise. Simulations show
that the membrane potential exhibits time symmetry. If the threshold is
neglected it is possible to approximate the response-stimulus correlation for
the sub-threshold regime. Starting with the potential at the threshold value
- where it has just produced a response spike - the flow of the potential can
be followed backwards in time. From this an explicit expression can be
derived which states the expected value of the driving stimulus at a given
time before the occuring response spike. All our results are compared to
numerical simulations of the corresponding stochastic processes to
demonstrate the quality of the approximations. The incorporation of more
biologically realistic parameters (like a more realistic noise model) is
subject of current investigation. Supported by Wellcome Trust
(061113/Z/00)
|
