A Theoretical Overview - Computational Neuroscience: A Comprehensive Approach - page 7

Synchronization The properties of the Kuramoto model have been intensively
studied in the literature [69]. Let us first introduce two parameters which characterize
the synchronization of a group of oscillators

Â
i

1
N

r ( t ) exp ( i y )= exp ( i q i ) (1.7)

= 1

Geometrically r is the order parameter describing the synchronization among neu-

rons. Numerically, it is shown that when the coupling strength K between neurons
is smaller than a constant K c , the neurons act as if they were uncoupled. r

( t )

( t ) decays

/ √ N .Butwhen K exceeds K c , the incoherent state

to a tiny jitter of size of order 1

becomes unstable and r grows exponentially, reflecting the nucleation of a small

cluster of neurons that are mutually synchronized, thereby generating a collective
oscillation. Eventually r

( t )

saturates at some level being smaller than 1. For some
most recent results on the Kuramoto model, we refer the reader to [54].
A detailed computation on how two neurons synchronize their activity has also
been carried out in [86]. They found that inhibitory rather than excitatory synaptic
interactions can synchronize neuron activity.

( t )

1.3 Stochastic dynamical systems

1.3.1 Jump processes

The observed electrical potentials of neurons as determined either by extracellular
or intracellular recording are never constant. The same is true for grossly recorded
field potentials and brain recordings such as the electroencephalogram. Often such
recordings of potential exhibit quite sudden changes or jumps. If the sample paths
of a continuous time random process have discontinuities then it is called a jump
process . A process may be a pure jump process, like a Poisson process or a random
walk, or there may be drift and or diffusion between the jumps.
Motivation for using jump processes in neurobiological modelling sprang primar-
ily from observations on excitatory and inhibitory synaptic potentials (EPSPs and
IPSPs). Examination of, for example, motoneuron or pyramidal cell somatically
recorded EPSPs may show a rapid depolarization of several millivolts relative to
resting potential, followed by an exponential decay with a characteristic time con-
stant [70]. A complete understanding of these events requires the use of complex
spatial models , (see below) but in the majority of studies attempting to model neu-
ronal electrophysiological properties in the last several decades, spatial extent has,
regrettably, been ignored, probably because of the unwillingness of but a few theo-
rists to confront partial differential equations rather than ordinary ones.
Putting aside the matter of spatial versus point models ,if N

is a
simple standard (unit jumps) Poisson process, with rate parameter l,thenasimple

= { N ( t ) , t ≥ 0 }