Model%20Reduction%20Techniques%20in%20Neuronal%20Simulation

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Model Reduction Techniques in Neuronal
Simulation Richard Hall, Jay Raol and Steven J.
Cox
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 ODEs. Applying
balanced model reduction to linear compartmental
fiber models can greatly reduce the complexity of
the problem.

• 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 PDEs to ODEs
• Simulation of many neurons can be a computational
  expensive task

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 (IntegrateFire 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, ?(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 ODEs

• HSVs describe the difficulty to reach and
  observe a state.
• Small HSVs relative to other ones can be
  truncated producing little error.

• Balanced model reduction
• Reduced of ODEs
• Maintained acceptable levels of error
• Applicable to linearized active fibers

• This system can be reduced from 159 to 2 state
  variables, with error 1.

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.
2. Antoulas, A. and Sorensen, D. Approximation of
   Large-Scale Dynamical Systems. Int. J. Appl.
   Math.Comput. Sci., 2001, 11(5), 1093-1121.