Shot Noise, diffusion limits and suitable approximations
Sprache des Vortragstitels:
5th International Conference on Mathematical NeuroScience (ICMNS)
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Suppose that events (e.g., jumps representing the excitatory inputs impinging on a neuron) occur in accordance to a Poisson process N(t) with constant rate lambda> 0. Associated with the i-th event is a positive random variable J_i, which quantifies the event (e.g., its amplitude). Denote by tau_i the time of the i-th event. When the J_i are independent and identically distributed random variables, independent of the Poisson process N(t), the underlying process X(t), t>=0 is called shot noise
process. We study the shot noise process (1) from two different perspectives. First, we consider different distributions for the jump amplitudes J_i and calculate the corresponding first three moments and stationary distribution of the shot noise process X(t). The obtained results are compared with the corresponding statistics of some well known process, e.g., the Ornstein-Uhlenbeck and the Feller processes, to investigate under which conditions these diffusion processes can satisfactorily approximate the shot noise X(t). Second, we consider a sequence of shot noise processes X_n(t),n>=1, with jump amplitudes going to 0 and jump frequencies going to infinity as n goes to infinity. Under which conditions does the sequence of shot noise processes converge to a diffusion process Y (t)? Is there any shot noise process fulfilling these conditions? If not, what type of limiting process Y(t) do we obtain? How is the distribution of J_i;n changing the properties of Y(t)?
We will discuss our results in the framework of neuronal models, and in particular Leaky Integrate-and-Fire models, where shot noises are used to model synaptic input currents and diffusion approximations are commonly considered to replace the shot noise input with a Gaussian white noise.
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