Similar IF neurons synchronize

[Walter Senn]

The following paper is accepted at the SIAM Journal on Applied Mathematics:

"Similar non-leaky integrate-and-fire neurons with instantaneous couplings
always synchronize"

The paper reconsiders the dynamics of pulse-coupled integrate-and-fire (IF)
neurons
analyzed by Mirollo and Strogatz (SIAM J.  Appl.  Math., 1990). Lifting their
restriction to identical oscillators, we study the case of different intrinsic
frequencies and different thresholds of the neurons, as well as different but
positive couplings. For non-leaky neurons, we prove that generically the
dynamics becomes fully synchronous for any initial conditions if the intrinsic
frequencies, the thresholds and the couplings are not too different. For the
case of non-leaky IF neurons, this confirms Peskin's conjecture (1975)
according to which nearly identical pulse-coupled oscillators in general
synchronize. In particular, leakyness, or more general, concave evolution
functions as imposed by Mirollo and Strogatz, are not necessary to insure
global synchronization. Our result differs from the findings of Mirollo and
Strogatz in that, for non-leaky IF neurons, almost all networks with weak
homogeneity converge for all initial conditions to synchronous firing, while
in their work with leaky and identical IF neurons, the dynamics synchronizes
for all set of parameter values, but for each set only for almost all initial
conditions. Interestingly, for non-leaky neurons the case of exactly identical
oscillators and weights is exceptional in that it does not assure full
synchronization for all initial conditions.

Walter Senn and Robert Urbanczik

The paper can be downloaded from http://www.cns.unibe.ch/~wsenn/#pub .


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Walter Senn                   Phone office: +41 31 631 87 21
Physiological Institute       Phone home:   +41 31 332 38 31 
University of Bern            Fax: +41 31 631 46 11
Buehlplatz 5                  email: wsenn at cns.unibe.ch
CH-3012 Bern SWITZERLAND      http://www.cns.unibe.ch/
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