Stochastic Differential Equation Model for Cerebellar Granule Cell Excitability

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.Author Summary: Computational modeling is of importance in striving to understand the complex dynamic behavior of a neuron. In neuronal modeling, the function of the neuron's components, including the cell membrane and voltage-dependent ion channels, is typically described using deterministic ordinary differential equations that always provide the same model output when repeating computer simulations with fixed model parameter values. It is well known, however, that the behavior of neurons and voltage-dependent ion channels is stochastic in nature. A stochastic modeling approach based on probabilistically describing the transition rates of ion channels has therefore gained interest due to its ability to produce more accurate results than the deterministic approaches. These Markov chain type of models are, however, relatively time-consuming to simulate. Thus it is important to develop new modeling and simulation approaches that take into account the stochasticity inherently present in the function of ion channels. In this study, we seek new stochastic methods for modeling the dynamic behavior of neurons. We apply stochastic differential equations (SDEs) and Brownian motion that are also commonly used in the air space industry and in economics. An SDE is a differential equation in which one or more of the terms of the mathematical equation are stochastic processes. Computer simulations show that the irregular firing behavior of a small neuron, in our case the cerebellar granule cell, is reproduced more accurately in comparison to previous deterministic models. Furthermore, the computation is performed in a relatively fast manner compared to previous stochastic approaches. Additionally, the SDE method provides more sophisticated mathematical analysis tools compared to other, similar kinds of stochastic approaches. In the future, the new SDE model of the cerebellar granule cell can be used in studying the emergent behavior of cerebellar neural network circuitry.