A Newton-based shooting method to find synaptic threshold in active cables
© Kath 2015
Published: 18 December 2015
The integration of synaptic inputs in a neuron can be nonlinear not just at the axon, but also locally in the dendrites if they are imbued with active voltage-gated ion channels. For example, CA1 pyramidal neurons have high densities of sodium and potassium and currents in their dendrites, and these densities can vary substantially in the arbor [1, 2]. Such nonlinearities can lead to compartmentalized responses to inputs , with branches acting as individual nonlinear units in which dendritic spikes occur. A cell can thus function as a multi-layered network with the soma as final output. This motivates determining when a given set of synaptic inputs is large enough to generate a local dendritic spike, or, alternatively, determining the synaptic conductance value(s) at threshold for producing a spike.
Here u represents the voltage and any gating variables. The first term on the right models the diffusive part of the cable equation, and the next term the nonlinearities from any active voltage-gated ion channels. The last term represents synaptic conductances at points in the dendritic tree with overall strength g s . An unstable threshold solution and accompanying synaptic strength can be found using 1) a preconditioned version of the steady-state cable equation combined with constraints requiring the difference between a shooting solution from rest, u(T,x;g s ), and the critical surface to be perpendicular to the single unstable eigenvector associated with the critical surface. The overall procedure finds the value of g s leading to a solution that asymptotes to the critical surface as t goes to ∞.
WLK was supported by NIH grant 1R01NS077601 & the Janelia Research Campus Visiting Scientist Program
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