A spatial stochastic neuronal model with Ornstein-Uhlenbeck input current

Abstract

We consider a spatial neuron model in which the membrane potential satisfies a linear cable equation with an input current which is a dynamical random process of the Ornstein-Uhlenbeck (OU) type. This form of current may represent an approximation to that resulting from the random opening and closing of ion channels on a neuron's surface or to randomly occurring synaptic input currents with exponential decay. We compare the results for the case of an OU input with those for a purely white-noise-driven cable model. The statistical properties, including mean, variance and covariance of the voltage response to an OU process input in the absence of a threshold are determined analytically. The mean and the variance are calculated as a function of time for various synaptic input locations and for values of the ratio of the time constant of decay of the input current to the time constant of decay of the membrane voltage in the physiological range for real neurons. The limiting case of a white-noise input current is obtained as the correlation time of the OU process approaches zero. The results obtained with an OU input current can be substantially different from those in the white-noise case. Using simulation of the terms in the series representation for the solution, we estimate the interspike interval distribution for various parameter values, and determine the effects of the introduction of correlation in the synaptic input stochastic process.

Characterization of Subthreshold Voltage Fluctuations in Neuronal Membranes

“…These models cover a broad range of complexity, ranging from (leaky) integrate-and-re neuron models (Lapicque, 1907) with gaussian white noise (Lánský & Lánská, 1987;Bindman, Meyer, & Prince, 1988;Tuckwell, 1988;Lánský & Rospars, 1995;Doiron et al, 2000;van Rossum, 2001;Brunel, Chance, Fourcaud, & Abbott, 2001), to stochastic membrane equations (Tuckwell & Walsh, 1983;Tuckwell, Wan, & Wong, 1984;Manwani & Koch, 1999a, 1999bTuckwell, Wan, & Rospars, 2002), up to biophysical-faithful models of single neurons in which synaptic background activity is incorporated by the random release at individual synaptic terminals according to Poisson processes (Bernander et al, 1991;Rapp et al, 1992;Lánský & Rodriguez, 1999;Manwani & Koch, 1999a, 1999bTiesinga et al, 2000;Rudolph & Destexhe, 2001a, 2001bRudolph et al, 2001;Tuckwell et al, 2002). As it was shown in Ricciardi and Sacerdote (1979), under point-like excitatory and inhibitory synaptic inputs Poisson-distributed in time, the neuron's membrane potential undergoes a continuous random walk.…”

“…However, the complexity of the resulting stochastic equations allows us to address only speci c problems analytically (e.g., statistical characteristics like mean and variance of the membrane potential; see Manwani & Koch, 1999a, 1999bTuckwell et al, 2002), whereas numerical methods (for a review, see Werner & Drummond, 1997) or approximations (e.g., meaneld approximation; see Van den Broeck, Parrondo, Armero, & Hernández-Machado, 1994;Ibañes, García-Ojalvo, Toral, & Sancho, 1999;Genovese & Muñoz, 1999) remain the standard tools. In addition, the particular mathematical form of noise terms and their incorporation into stochastic neuron models (i.e., the nature of the coupling of the noise to the neural system) are still the subject of controversy.…”

Inferring network activity from synaptic noise

“…By combining intracellular recordings with biophysically detailed computational models of single neurons, it was possible to estimate parameters characterizing synaptic activity (such as synaptic densities, quantal conductances and release rates) and, this way, to capture some of the properties characterizing synaptic noise [22,67]. In such models, synaptic background activity is commonly described by thousands of individual synaptic terminals which release randomly according to Poisson processes [5,22,43,52,58,59,70,75,77,96,100]. Here, conductance changes during release at single synaptic terminals are described by kinetic models of postsynaptic receptors for excitatory (glutamate AMPA and NMDA receptors) and inhibitory (GABAergic receptors; Fig.…”

“…Such models cover a broad range of complexity, starting with integrateand-fire models [7,11,28,53,54,98,104], stochastic membrane equations [51,58,[99][100][101], up to biophysically detailed models of single neurons with release activity at thousands of distributed individual synaptic terminals, each described by Poisson processes [5,22,43,52,70,[75][76][77]96,100]. An intermediate approach is to model synaptic noise with two global conductances described by stochastic processes [27].…”

On modeling the neuronal activity in movement disorder patients by using the Ornstein Uhlenbeck Process

“…The OUP is a modified Wiener Process based on leaky integration assumption [3] that can model the randomness of the Inter Spike Interval (ISI), the time interval between two consecutive neuronal firings. It captures the spike generation mechanism, which is ignored in a simplistic model like the Poisson Process (PP), and also accounts for the change in membrane potential between two firing events, unlike in Brownian Motion (BM) models [4].…”