Neurophysics

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

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

3. 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 • An action potential is produced in the soma • 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 ?

4. The longitudinal resistance of an axon or dendrite is: with rL - 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)

5. 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

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

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

8. Under the assumption that rL 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 im

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

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

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

12. injected current through electrode surface area of compartment specific membrane capacitance (Fm2) membrane currents due to ion-channels / membrane area • The membrane potential dynamics of a single isolated compartment is described by: • Several compartments coupled in a non-branching manner:

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

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