Neuroelectronics The Neuron Neuron The Device Synapse Dendrites

Neuroelectronics

The Neuron

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

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

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

The Membrane: Capacitance Current I Outside Voltage V Inside C = Q/V (Q = CV); d. Q/dt = I I = C d. V/dt 1 Farad = 1 Coulomb/ 1 Volt V I

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

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

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

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

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

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

Passive membrane: Compartmental Model

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

Active membrane: Sodium Channel

Active membrane: Voltage Dependent Conductance Na+ Channels: GNa (1/RNa) and ENa K+ Channels: GK (1/RK) and EK Ca 2+ Channels: GCa (1/RCa) and ECa Leak Channels: GL (1/RL) and EL

Active membrane: Hodgkin Huxley Equations I=Cd. V/dt+GL(V-EL)+GKn 4(V-EK)+GNam 3 h(V-ENa)

Active membrane: Hodgkin Huxley Equations dn/dt=an(V)(1 -n)-bn(V)n an(V) = opening rate bn(V) = closing rate dm/dt=am(V)(1 -m)-bm(V)m am(V) = opening rate bm(V) = closing rate dh/dt=ah(V)(1 -h)-bh(V)h ah(V) = opening rate bh(V) = closing rate an=(0. 01(V+55))/(1 -exp(-0. 1(V+55))) bn=0. 125 exp(-0. 0125(V+65)) am=(0. 1(V+40))/(1 -exp(-0. 1(V+40))) bm=4. 00 exp(-0. 0556(V+65)) ah=0. 07 exp(-0. 05(V+65)) bh=1. 0/(1+exp(-0. 1(V+35)))

Active membrane: Synaptic Conductance Synaptic Channels: GSyn (1/RSyn) and ESyn

Reduced Model: Leaky Integrate and Fire Cd. V/dt = -GL(V-EL) + 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 PSP’s • 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

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

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