Transcript VIGRE.Fall06.ModelRe..

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:
1. State-space Representation
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.
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
•A simple model of neural fibers uses an RLC
circuit for each compartment, resulting in a system
of ODE’s
states which are hardest to reach/observe
3. 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 .
To express this mathematically, consider the following:
• A neuron in the small network is modeled via an ODE (Integrate&Fire Neurons)
•HSV’s describe the difficulty to reach and observe
a state.
•Small HSV’s relative to other ones can be
truncated producing little error.
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.
•Neurons have the same passive properties (i.e. membrane time constant,
inhibitory/excitatory conductance) and are sparsely connected within themselves
Balanced model reduction
•Reduced # of ODE’s
• All neurons receive the same input from neurons outside their own network
•Maintained acceptable levels of error
These assumptions give rise to a PDE describing the probability density, ρ(t,v)
•Applicable to linearized active fibers
•This system can be reduced from 159 to 2 state
variables, with error ~1%.