Neuroelectronics

The Neuron

Input Output Threshold Time SynapseSynapse Dendrites (Input)Dendrites (Input) Cell BodyCell Body Axon (Output)Axon (Output) Neuron: The Device Equilibrium: Membrane Potential Dendrites: Passive Conductance Axon: Spikes (Hodgkin Huxley Eqns)

Approach Basic Concepts Detailed Model of Neuron Reduced Model of Neuron Network Model

The Membrane Membrane: 3 to 4 nm thick, essentially impermeable Ionic Channels: Selectively permeable (10,000 times smaller resistance)

The Membrane: Capacitance Current I C = Q/V 1 Farad = 1 Coulomb/ 1 Volt (Q = CV); dQ/dt = I I = C dV/dt Outside InsideI V Voltage V

The Membrane: Resistance Current I R = V/I 1 Ohm = 1 Volt/ 1 Ampere I = V/R Outside Inside I V Voltage V

The Membrane: Capacitance and Resistance Current I I = C dV/dt + V/R (C*R) dV/dt = -V + IR Outside Inside I V Voltage V

The Membrane: Membrane Potential Case 1: Single type of Ion (Na + ) Charge Balanced out by impermeable ion Owing to Concentration gradient Owing to Voltage gradient Reversal Potential Reversal Potential : When opposing currents balance each other out. Nernst Equation: E= (RT/z) ln([outside]/[inside]) Reversal Potential for Na + is around +50 mV (based on typical concentration gradients) Note: Reversal potential does not depend upon resistance. Outside Inside

The Membrane: Membrane Potential Case 1: Two types of Ions (Na + and K + ) Equilibrium Potential Equilibrium Potential : When opposing currents balance each other out ( -70 mV). Goldman Equation: V= (-60 mV)* log 10 ((P K [K] in +P Na [Na] in +P Cl [Cl] out )/(P K [K] out +P Na [Na] out +P Cl [Cl] in )) Note: Equilibrium potential does depend upon relative resistances. Reversal potentials ---- Na + : +50 mV K + : -80 mV Outside Inside Why ingesting Pottasium Cloride is deadly; ingesting Sodium Cloride is not.

Passive membrane: Equivalent Circuit Voltage independent channels Single Compartment Electrotonically compact neuron. I INJ = I I = C dV/dt + (V-E L )/R Use new variable: V = V - E L (C*R) dV/dt = -V + IR Inside Outside

Passive membrane: Cable Equation Voltage independent channels Multiple Compartments Electrotonically non-compact neuron. C V/t = -V/R + I V/x = ir hence 2 V/x 2 = ri/x I INJ = I - i/x hence I = I INJ + i/x CV/t = (1/r) 2 V/x 2 – (1/R)V + I INJ Inside Outside

Passive membrane: Compartmental Model

Active membrane: Voltage Dependent Conductance Na Channel Activate Inactivate Deactivate DeInactivate K Channel Activate Deactivate Outside Inside

Active membrane: Sodium Channel

Active membrane: Voltage Dependent Conductance Na + Channels: G Na (1/R Na ) and E Na K + Channels: G K (1/R K ) and E K Ca 2+ Channels: G Ca (1/R Ca ) and E Ca Leak Channels: G L (1/R L ) and E L

Active membrane: Hodgkin Huxley Equations I=CdV/dt+G L (V-E L )+G K n 4 (V-E K )+G Na m 3 h(V-E Na ) I=CdV/dt+G L (V-E L )

Active membrane: Hodgkin Huxley Equations dn/dt=a n (V)(1-n)-b n (V)n a n (V) = opening rate b n (V) = closing rate dm/dt=a m (V)(1-m)-b m (V)m a m (V) = opening rate b m (V) = closing rate dh/dt=a h (V)(1-h)-b h (V)h a h (V) = opening rate b h (V) = closing rate a n =(0.01(V+55))/(1-exp(-0.1(V+55))) b n =0.125exp( (V+65)) a m =(0.1(V+40))/(1-exp(-0.1(V+40))) b m =4.00exp( (V+65)) a h =0.07exp(-0.05(V+65)) b h =1.0/(1+exp(-0.1(V+35)))

Active membrane: Synaptic Conductance Synaptic Channels: G Syn (1/R Syn ) and E Syn

Reduced Model: Leaky Integrate and Fire CdV/dt = -G L (V-E L ) + I Assume that synaptic response is an injected current rather than a change in conductance. Assume injected current is a δ function; Results in PSP Linear System: Total effect at soma = sum of individual PSPs Neuron Spikes when total potential at soma crosses a threshold. Reset membrane potential to a reset potential (can be resting potential)

Network Models Biggest Difficulty: Spikes Membrane Potential Spikes Membrane Potential Spikes Membrane Potential

Firing Rate Model Exact spike sequence converted into instantaneous rate r(t) Justification: Each neuron has large number of inputs which are generally not very correlated. 2 Steps: Firing Rate of Presynaptic Neuron Synaptic Input to Postsynaptic Neurons Total Input to Postsynaptic Neuron Firing rate of Postsynaptic Neuron Total Synaptic Input modeled as total current injected into the soma f-I curve: Output Spike Frequency vs. Injected Current curve

Firing Rate Model Firing rate does not follow changes in total synaptic current instantaneously, hence τ dv/dt = -v +F(I(t)) I(t)=w.u(t) Input u Output v

Firing Rate Network Model τ dv/dt = -v + F( w.u(t)) v(t) u(t)