Nonstationary filtered shot-noise processes and applications to neuronal membranes 

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State of the Art and Main Contributions

The theoretical basis from this work can be found in two important developments: the basic membrane model for the integrate-and-fire neuron introduced by Lapicque [Lapicque, 1907, Abbott, 1999] and the discovery and modeling of shot noise by Campbell and Schottky [Campbell, 1909, Schottky, 1918].
Lapique’s membrane model describes deterministic current input and was later extended to include stochastic currents, conductances and other biophysical aspects [Stein, 1965, Verveen and DeFelice, 1974]. Further developments include non-linear membrane potential dynamics, synaptic plasticity and adaptation but are not addressed here [Tuckwell, 1988b,a, Dayan and Abbott, 2001, Gerstner et al., 2014].
Shot noise processes are simple yet powerful models of stochastic input that correspond to the superposition of impulse responses arriving at random times according to a Poisson law. The early works of Campbell and Schottky described current fluctuations in vacuum tubes but many applications were later found in biology [Stevens, 1972, Siebenga et al., 1973], acoustics [Kuno and Ikegaya, 1973], optics [Rousseau, 1971, Picinbono et al., 1970], wireless communications [Venkataraman et al., 2006] and many other fields [Snyder and Miller, 1991, Parzen, 1999]. Whereas Campbell derived the expressions for the stationary mean and variance of shot noise, in-depth analysis of their probabilistic structure was performed by S.O. Rice (in addition to many important properties of Gaussian processes) [Rice, 1944, 1945]. A modern review of more recent developments are presented in Refs. [Rice, 1977, Snyder and Miller, 1991, Parzen, 1999]. Shot noise has a simple mathematical form but can display nonstationary and non- Markovian characteristics: a time-varying rate of random arrival times yields nonstationary shot noise, and the process is in general non-Markovian for a single state variable [Masoliver, 1987, Lund et al., 1999], with a notable exception being the exponential kernel. The shot noise assumption is motivated by experimental studies showing that presynaptic spikes elicit stereotypical responses under low activity regimes [Hodgkin et al., 1952, Hodgkin and Huxley, 1952, Fatt and Katz, 1952, Curtis and Eccles, 1960]. This neglects changes in synaptic response due to synaptic saturation and short term plasticity that are certainly relevant but provides a reasonable first approximation.

Poisson Point Processes in the Real Line

Poisson point processes model the distribution of points in arbitrary dimensions. The coordinates of stars in a small section of the sky or the location of trees in a forest provide examples in two or three spatial dimensions. In one dimension they can provide a model for the generation of presynaptic spike times.
A PPP (S, λ) that generates points or event times in an interval S ⊆ R of the real line is characterized by a non-negative rate function λ(x) ≥ 0 such that the quantity m(S) ≡ R S λ(x) dx is finite for any bounded interval S. A PPP is said to be homogeneous if the rate function λ(t) = λ is constant and inhomogeneous otherwise.
A realization of the PPP is a set ξ ≡ {n ≥ 0, {x1, . . . , xn} ∈ S} that specifies the number of points n and their locations {x1, . . . , xn} ∈ S. These points are associated with presynaptic spike times of synaptic input. In order to simplify notations n is made implicit and ξ represents the event times in order to write more compact expressions such as P xj∈ g(t, xj).
A realization ξ is obtained through a two-step sampling procedure: an integer n ≥ 0 is drawn from a Poisson distribution with mean m(S); and for n > 0, each xj is independent and identically distributed (i.i.d.) with probability density p(xj) = λ(xj)/m(S). The condition of finite m(S) over bounded S ensures a finite number of event times for realizations ξ over bounded S. A well-known and simple implementation of this procedure is included in the Appendix A.1. Examples of PPP realizations for both homogeneous and inhomogeneous PPP are shown in Fig. 2.1.

Multiplicative Noise

The case of multiplicative noise corresponds to conductance synapses and is developed in the articles of Chapters 3 and 4. In particular, the case of a single synapse type can be expressed as a pure multiplicative noise process as shown in the article of Chapter 3. This leads to shorter expressions by avoiding the usage of Slivnyak-Mecke Theorem. The solution for the case of two independent shot noise inputs and a time dependent current is briefly derived here and the complete derivation for the case J(t) = 0 is provided in Appendix A.4. Comparisons with numerical simulations are shown in the relevant articles.

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Asymptotic and Stationary Limits

The statistical properties of long running shot noise process Qt and system response Yt are analyzed in this section. A shot noise process generated from an homogeneous PPP will reach a stationary regime after an initial transient period, assuming the appropriate convergence properties of the shot noise kernel to ensure the finiteness of cumulants (Eq. (2.13)). This may no longer be the case under inhomogeneous PPP. However, an asymptotic limit may exist for certain periodic rate functions such as the example from Fig. 2.9.
The stationary or asymptotic limits of shot noise process Qt are obtained by setting the origin of event arrivals T0 at infinity (T0 → −∞). The stationary and asymptotic limits of system response Yt are obtained by setting the start of input integration t0 in the same limit (t0 = T0, T0 → −∞). The filtering of Qt may eventually be decoupled from the starting time of event arrivals, as shown in the lower plots of the same Figure. Finally, the limit where Yt is driven by stationary or asymptotic Qt and the initial Qt transients are neglected, is obtained by setting T0 → −∞and keeping t0 finite.

Causal Point Process Transformations

We review the basic properties of PPP transformations and analyze the stochastic process generated by causal PPP transformations. The expectation of PPP transformations yields the joint cumulants of the associated processes. We illustrate this approach with the shot noise process and compare the predicted mean and second order cumulants with numerical simulations. We consider a PPP (S, λ) that generates points in the interval S ⊆ R of the real line with rate function λ(x) ≥ 0 such that m(S) ≡ R S λ(x) dx is finite for any bounded interval S. A realization ξ of contains a set of n ≥ 0 points {x1, . . . , xn} ∈ S that we associate with input arrival times. A PPP is said to be homogeneous for constant λ(t) = λ and inhomogeneous otherwise. Example rate functions and sample realizations of the associated inhomogeneous PPP are shown in Fig. 2. These rate functions were used to generate input arrival times for the filtered shot noise process of Sec. 2 and the presynaptic spikes for the neuronal membrane of Sec. 6.

Table of contents :

I Introduction
1 General Introduction 
1.1 Synopsis
1.2 Framework Description
1.3 Applications
1.4 State of the Art and Main Contributions
1.5 Thesis Outline
2 Analytical Tools 
2.1 Poisson Point Processes
2.1.1 Poisson Point Processes in the Real Line
2.1.2 PPP Transformations
2.1.3 Membrane Equation as PPP transformation
2.1.4 Statistics of PPP transformations
2.2 General Solution
2.2.1 Additive Noise
2.2.2 Multiplicative Noise
2.2.3 General Case
2.3 Asymptotic and Stationary Limits
2.4 Random Dirac Delta Sums
2.5 Compound PPP Transformations
2.6 Central Moments Expansion
Appendix A 
A.1 Sampling Procedure
A.2 Numerical Integration of Membrane Equation
A.3 Cumulants of Integral PPP Transformations
A.4 Two Independent Conductance Inputs
A.5 General Case
A.6 Random Dirac Delta Sums
A.7 Shot Noise Cumulants
A.8 Expectation of Random Product
II Research Articles 
3 Nonstationary filtered shot-noise processes and applications to neuronal membranes 
4 The impact of synaptic conductance inhomogeneities on membrane potential statistics
5 How causal correlations between synaptic inputs affect membrane potential fluctuations
6 Estimating stochastic process memory in neuronal membranes 
III Discussion 
7 General Discussion 


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