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Neurophysics Adrian Negrean - part 2 - adrian.negrean@cncr.vu.nl

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1. Aim of this class 2. A first order approximation of neuronal biophysics 1. Introduction 2. Electro-chemical properties of neurons 3. Ion channels and the Action Potential 4. The Hodgkin-Huxley model 5. The Cable equation 6. Multi-compartmental models Contents

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The Cable equation Describes the propagation of signals in electrical cables, and in this case it will be applied to dendrites and axons Case study: Simultaneous intracellular recordings from soma and dendrite A)An action potential is produced in the soma B)A set of axon fibers is stimulated to produce a compound excitatory post- synaptic potential What are the differences and how do you explain them ?

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The longitudinal resistance of an axon or dendrite is: withr L - intracellular resistivity ( m) Δx - segment length a - segment radius The intracellular resistivity depends on the ionic composition of the intracellular milieu (and on the distribution of organelles)

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The longitudinal current through such a segment is: where ΔV(x,t) is the voltage gradient across the segment Currents flowing in the increasing direction of x are defined to be positive

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In the limit : Besides the longitudinal currents, there are several membrane currents flowing in/out of the segment: do you understand the formula ?

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Applying the principle of charge conservation for the previous cable segment we get: Divide the above by such that the r.h.s. is in the limit

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Under the assumption that r L does not vary with position the cable equation is obtained: The radius of the cable is allowed to vary to simulate the tapering of dendrites Boundary conditions required for V(x,t) and Linear cable approximation: Ohmic membrane current i m

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Use change of variables And multiply by r m with membrane time constant to get : and electrotonic length (in the linear cable approximation)

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Steady state (A) and transient (B) solutions to the linear cable equation:

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Multi-compartmental models To calculate the membrane potential dynamics of a neuron, the cable equation has to be discretized and solved numerically

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The membrane potential dynamics of a single isolated compartment is described by: injected current through electrode surface area of compartment membrane currents due to ion-channels / membrane area specific membrane capacitance (Fm 2 ) Several compartments coupled in a non-branching manner:

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The Ohmic coupling constants between two compartments with same length and radii:

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Next time you see a neuron, you should see this:

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