608 research outputs found
Modeling networks of spiking neurons as interacting processes with memory of variable length
We consider a new class of non Markovian processes with a countable number of
interacting components, both in discrete and continuous time. Each component is
represented by a point process indicating if it has a spike or not at a given
time. The system evolves as follows. For each component, the rate (in
continuous time) or the probability (in discrete time) of having a spike
depends on the entire time evolution of the system since the last spike time of
the component. In discrete time this class of systems extends in a non trivial
way both Spitzer's interacting particle systems, which are Markovian, and
Rissanen's stochastic chains with memory of variable length which have finite
state space. In continuous time they can be seen as a kind of Rissanen's
variable length memory version of the class of self-exciting point processes
which are also called "Hawkes processes", however with infinitely many
components. These features make this class a good candidate to describe the
time evolution of networks of spiking neurons. In this article we present a
critical reader's guide to recent papers dealing with this class of models,
both in discrete and in continuous time. We briefly sketch results concerning
perfect simulation and existence issues, de-correlation between successive
interspike intervals, the longtime behavior of finite non-excited systems and
propagation of chaos in mean field systems
Neighborhood radius estimation in Variable-neighborhood Random Fields
We consider random fields defined by finite-region conditional probabilities
depending on a neighborhood of the region which changes with the boundary
conditions. To predict the symbols within any finite region it is necessary to
inspect a random number of neighborhood symbols which might change according to
the value of them. In analogy to the one dimensional setting we call these
neighborhood symbols the context of the region. This framework is a natural
extension, to d-dimensional fields, of the notion of variable-length Markov
chains introduced by Rissanen (1983) in his classical paper. We define an
algorithm to estimate the radius of the smallest ball containing the context
based on a realization of the field. We prove the consistency of this
estimator. Our proofs are constructive and yield explicit upper bounds for the
probability of wrong estimation of the radius of the context
Partially observed Markov random fields are variable neighborhood random fields
The present paper has two goals. First to present a natural example of a new
class of random fields which are the variable neighborhood random fields. The
example we consider is a partially observed nearest neighbor binary Markov
random field. The second goal is to establish sufficient conditions ensuring
that the variable neighborhoods are almost surely finite. We discuss the
relationship between the almost sure finiteness of the interaction
neighborhoods and the presence/absence of phase transition of the underlying
Markov random field. In the case where the underlying random field has no phase
transition we show that the finiteness of neighborhoods depends on a specific
relation between the noise level and the minimum values of the one-point
specification of the Markov random field. The case in which there is phase
transition is addressed in the frame of the ferromagnetic Ising model. We prove
that the existence of infinite interaction neighborhoods depends on the phase.Comment: To appear in Journal of Statistical Physic
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