BME 6938 Neurodynamics Instructor: Dr Sachin S. Talathi

Recap Fundamental laws of cellular neurophysiology Nernst Plank Equation Nernst Equation (Reversal potential for ion channel) Active and passive ion transport (and Donnan equilibrium potential) Constant Field Assumption Goldman Hodgkin Katz Model Goldman Voltage Equation

Ionic currents and conductances Current across cell membrane resulting from flow of ion X: GHK Current Eq. In equilibrium : V=E X, current vanishes, i.e., I X (E X )=0 Linearize I X (V) around E X ; we get from GHK Eq. conductance Reversal (Nernst) potential Driving force (in general function of V and t) Note: Convention +ve driving force=> Current flowing out of cell Note:

The Equivalent circuit model for cell membrane Total membrane current I, is given as sum of all ionic currents and the capacitive current caused by the bi- lipid layer capacitance Note Ordinary differential equation model for neuron carrying current via flow of Na +,K + and Cl - ions across its cell membrane At rest (Inward current) (Outward current)

Resting Potential and Input Resistance Resting potential V rest,corresponds to steady state conditions i.e., With total input conductance Input resistance measures the sensitivity of the asymptotic membrane potential to input current Define Note: In general g inp, R inp and depend on time and voltage

Integrate and Fire (IF) model for nerve cell (Lapicque model) Assume V rest =0 and membrane resistance R inp does not depend on time and voltage General Solution: where And H(t) is the Heaviside step function G(t) is the Greens function Models the subthreshold dynamics of cell membrane voltage

Examples of sub threshold behavior Example 1: Example 2:Alpha function

IF model with threshold As we saw earlier, the IF model is only appropriate for subthreshold responses In order for IF model to mimic neuron that produces action potential, a threshold condition needs to be superimposed. Spiking IF model is complete by assuming that the nerve cell generates an action potential everytime V(t) reaches threshold levels. A simple threshold condition is t i is the sequence of times when spikes occur and t R is the absolute refractory period

Sub threshold repetitive excitation What is the minimum frequency of repetitive stimulation necessary to make the cell generate action potential? Assume ; sequence of delta pulse impulses with strength k and pulse freq 1/T From Greens function analysis, we know that every time a pulse arrives, the voltage will rise to k and then decay exponential per In steady state if V max and V min are the range of subthreshold voltage we have For neuron to spike

Exercise Derive expression for T crit when Hint

Frequency Current Relationship (Gain Function) Super threshold excitation Rheobase current I Rh : The membrane current at which the cell reaches threshold potential Change of variables: IF model follows linear F-I curve

Graphical analysis of IF neuron dynamics Subthreshold membrane activity Superthreshold membrane activity

Nonlinear IF model Extension to the IF model have been proposed to allow neurons to exhibit richer dynamics (i.e. bistability; we will study this in details at later date) Quadratic integrate and fire Exponential integrate and fire Subthreshold Dynamics: Superthreshold Dynamics:

Comparison to real neural dynamics in presence of noisy input current Trocme et al, 2003