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Model Reduction Techniques in Neuronal Simulation Richard Hall, Jay Raol and Steven J. Cox

Model Reduction Techniques in Neuronal Simulation Richard Hall, Jay Raol and Steven J. Cox. Why are model reduction methods necessary? There are over 10 11 neurons and 10 14 synapses in the Human brain An individual neuron can be modeled in many different ways from PDE’s to ODE’s

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Model Reduction Techniques in Neuronal Simulation Richard Hall, Jay Raol and Steven J. Cox

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  1. Model Reduction Techniques in Neuronal Simulation Richard Hall, Jay Raol and Steven J. Cox • Why are model reduction methods necessary? • There are over 1011 neurons and 1014 synapses in the Human brain • An individual neuron can be modeled in many different ways from PDE’s to ODE’s • Simulation of many neurons can be a computational expensive task Balanced Model Reduction of Neural Fibers Models of neural fibers require the spatial discretization of the fibers into smaller compartments. These multi-compartment models result in a system of linear ODE’s. Applying balanced model reduction to linear compartmental fiber models can greatly reduce the complexity of the problem. Method: Application: • Population Density Models • There exist neurons in small networks with similar properties that can be grouped together in one functional group. In addition, it is often the case that individual voltages for the neurons do not need to be calculated, only the average group response. Therefore, only the probability density of voltage and time is calculated for all the neurons in the group. • To express this mathematically, consider the following: • A neuron in the small network is modeled via an ODE (Integrate&Fire Neurons) • Neurons have the same passive properties (i.e. membrane time constant, inhibitory/excitatory conductance) and are sparsely connected within themselves • All neurons receive the same input from neurons outside their own network • These assumptions give rise to a PDE describing the probability density, ρ(t,v) • Initial numerical results indicate up to a 600x speed up versus the full ode system • A simple model of neural fibers uses an RLC circuit for each compartment, resulting in a system of ODE’s • HSV’s describe the difficulty to reach and observe a state. • Small HSV’s relative to other ones can be truncated producing little error. • Balanced model reduction • Reduced # of ODE’s • Maintained acceptable levels of error • Applicable to linearized active fibers • This system can be reduced from 159 to 2 state variables, with error ~1%. • Nykamp, D. and Tranchina, D. A Population Density Approach That Facilitates Large-Scale Modeling of Neuronal Networks. J. Comp. Neuro., 2000, 8, 19-50. • Antoulas, A. and Sorensen, D. Approximation of Large-Scale Dynamical Systems. Int. J. Appl. Math.Comput. Sci., 2001, 11(5), 1093-1121.

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