Simona Olmi - CNR ISC Firenze	#
Stability of the splay state in networks of pulse-coupled neurons #

The stability of the dynamical states characterised by a uniform 
firing rate
("splay states") is analysed in a network of N globally 
pulse-coupled
rotators (neurons) subjected to a generic velocity field. This is 
done by reducing the set of differential equations to an event 
driven map that is investigated in the limit of large network size.
We show that the Floquet spectrum characterising the stability of 
the
splay state can be decomposed in two components: (i) a 
long-wavelength component
which is the only one usually considered in mean-field analysis
[Abbott-van Vreesvijk, 1993]; (ii) a short-wavelenght component 
measuring the instability of ``finite-frequency" modes.
By developing a perturbative technique, we have found analytically
that, in the limit of large N, the short-wavelenght spectrum scales 
as 1/N^2 for generic discontinuous velocity fields.
Moreover, the stability of this component is determined by the sign 
of
the jump at the discontinuity. Altogether, the form of the spectrum 
depends on the pulse shape but is independent of the velocity 
field.
Furthermore, numerical results indicate that in the case of 
continuous
velocity fields, the Floquet exponents scale faster than 1/N^2
(namely, as 1/N^4) and we even find strictly neutral directions in 
a wider class than the sinusoidal velocity fields considered by 
Watanabe and Strogatz in Physica D 74 (1994) 197-253.
<br/>
<br/>
References
<br/>
R. Zillmer, R. Livi, A. Politi, and A. Torcini, Phys. Rev. E 76 
(2007)
046102
M. Calamai, A. Politi, and A. Torcini,  Phys. Rev. E 80, 036209 
(2009)
<br/>
S. Olmi, A.Politi, and A. Torcini,
"Stability of the splay state in networks of pulse-coupled 
neurons", submitted to J. Mathematical Neuroscience (2012)