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 1.State-space Representation Balanced model reduction is based on the A,B,C, and D matrices of a state-space representation 2. Compute the Reachability (P) and Observability (Q) Gramians The eigenvectors corresponding to the smallest eigenvalues of P/ Q indicate the states which are hardest to reach/observe 1.Transform the system to a Balanced Representation A transform based on P and Q can be found which will “balance” the system. The P and Q matrices of a “balanced” system will be equivalent, and their entries are called Hankel singular values. 4. Determine size of reduced system and Truncate The relative decay of the entries in the P/Q matrix indicates the amount of error that will result from the reduction. 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. A simple model of neural fibers uses an RLC circuit for each compartment, resulting in a system of ODE’s Method: Application: HSV’s describe the difficulty to reach and observe a state. Small HSV’s relative to other ones can be truncated producing little error. This system can be reduced from 159 to 2 state variables, with error ~1%. Balanced model reduction Reduced # of ODE’s Maintained acceptable levels of error Applicable to linearized active fibers Why are model reduction methods necessary? There are over neurons and 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 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 1.Nykamp, D. and Tranchina, D. A Population Density Approach That Facilitates Large-Scale Modeling of Neuronal Networks. J. Comp. Neuro., 2000, 8, Antoulas, A. and Sorensen, D. Approximation of Large-Scale Dynamical Systems. Int. J. Appl. Math.Comput. Sci., 2001, 11(5),