Electrical properties of neurons Rub é n Moreno-Bote

1. Electrical properties of neuronsRubén Moreno-Bote

2. Galvani frog’s legs experiment

3. Overview: • Passive properties of neurons (resting potential) • Action potential (generation and propagation). • Synaptic currents (AMPA, GABA, NMDA). • 4. Reduced models of neurons (LIF, QIF, LNP). • 5. Neuronal networks (balance and chaos).

4. 1. Passive properties of the neuron membrane 1.1 Membrane potential. Outside Inside Current, I Vout Vin Vin-Vext = -RI Ohm’s law I>0, inward current, which means that Vin is negative, Vin = -70 mV (we define Vout=0) I, current-> Amperes, A=C/s (order of magnitude: 10 nA, 10 µA) V, potential-> Volts (100mV, action potential; 0.1-1mV, postsynaptic current) R, resistance-> Ohms, , V/A (1 M) g, conductance, g=1/R-> Siemens, S (1/ ) (order: µS). Ohm’s law with conductances: I=-g(Vin-Vout)

5. Outside Inside 1.1 Membrane potential. Currents, resistances and capacitors. Current, I Vout capacitor Vin Membrane is impermeable to ions and creates the voltage difference (=Capacitor). Q=CV C, Capacitance-> Faradays (order 1 nF) Extra and intracellular fluid is electrically neutral.

6. Outside Inside 1.2 Ions and Ion Channels [Ca2+] [Cl-] [Na+] [K+] [K+] [Na+] [Ca2+] [Cl-] Cations: + Anions: - Channels are selective to particular ions. Passive vs. Active channels. Permeability is very high to K and Na, medium to Cl and very low to big anions. [K+]in=20 [K+]out [Na+]out=10 [Na+]in Main question: How is the membrane potential is related to [charges] in and out?

7. Outside Inside 1.3 Equilibrium potential for one ion • Two competing forces: • Diffusion by concentration gradient. • Motion by voltage gradient. [K+] [K+] Diffusion Voltage difference

8. 1.3 Equilibrium potential for one ion Diffusion Flux: Jdif(x)= -D d[K+](x) / dx D, diffusion coefficient-> D=µkT/q µ, mobility k, Boltzmann constant q, ion charge [K+] x [K+] Voltage difference Flux: Jelec(x)= -µz [K+](x) dV(x) / dx µ, mobility z, ion valence, +/-1, +/-2, etc. Equilibrium happens when Jdif(x) + Jelec(x) = 0, which leads to the Nernst equation: E K+ = kT / zq ln [K+]out / [K+]in = -75,-90 mV E K+ is the potential necessary to maintain the concentration gradient [K+]out / [K+]in

9. 1.3 Equilibrium potential for one ion [K+] E K+ = kT / zq ln [K+]out / [K+]in = -75,-90 mV E Na+ = +55 mV E Ca2+ = +150 mV E Cl- = -60,-65mV x [K+] Na+ K+ K+ Na+ Vm = -70 mV E K+ < Vm < E Na+ Compensated by Na-K pump.

10. 1.4 Equilibrium potential with K and Na channels Equilibrium: I K+ + I Na+ = 0 g K+(Vm-E K+) + g Na+(Vm-E Na+) = 0 g K+ g Na+ EL = Vm = (g K+E K+ + g Na+E Na+) / (g K++ g Na+) EL = -69 mV I K+ I Na+ + - E K+ E Na+ - + Leak Current: IL = I K+ + I Na+ Vm IL = g L (Vm - EL) g L = g K++ g Na+

11. V I 1.5 RC circuit for the passive membrane Leak Current: IL = g L (V m - EL) Capacitor Current: IC = C dV m /dt ( Q = C V ) g L + + + I L I C C + - - - External Stimulation: IC + IL = Iext(t) E L - RC passive membrane equation: C dVm / dt = -gL (V m - EL) + Iext(t) m = C / gL = 20nF / 1µS = 20ms Iext(t) Vm Vm

12. 2 Action Potential 2.1. Active ion channels. Active membrane P, prob of being active can depend on several factors. Active Channels: -Voltage-gated (Na, K, etc) -Extracellular ligand gated (e.g. synaptic receptors) -Intracellular ligand gated (e.g. Ca-depenent channel) 5 pA 500 ms Patch-clamp technique (E. Neher and B. Sakmann, 1976)

13. Outside Inside 2.1 Active ion channels K+ Vm = -70 mV E K+ < Vm < E Na+ Voltage-clamp Voltage-dependent K+ channel (Persistent)

14. Outside Inside 2.1 Active ion channels Na+ Vm = -70 mV E K+ < Vm < E Na+ Voltage-dependent Na+ channel (Transient)

15. 2.2 Dynamics of ion channels n4 is the probability that the potassium channel is open m3h is the probability that the sodium channel is open activation gates inactivation gates αis the probability a closed gate will open βis the probability an open gate will close (V) Close Open (V)

16. 2.3 Hodgkin and Huxley equations Na+ Channels: GNa (1/RNa) and ENa=55mV K+ Channels: GK (1/RK) and EK=-80mV Ca2+ Channels: GCa (1/RCa) and ECa Leak Channels: GL (1/RL) and EL=-70mV

17. 2.3 Hodgkin and Huxley equations Time const. Steady State Spike Generation: Iapp ↑ → V↑ → m↑ (quickly) while n↑ and h↓ (slowly) Thus V goes up quickly toward ENa until h shuts off Na channels and K inhibition dominates

18. 2.3 Hodgkin and 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=0.125exp(-0.0125(V+65)) am=(0.1(V+40))/(1-exp(-0.1(V+40))) bm=4.00exp(-0.0556(V+65)) ah=0.07exp(-0.05(V+65)) bh=1.0/(1+exp(-0.1(V+35)))

19. 2.4 Spatially distributed neuron models point neuron model The spatial distribution of ion channels is almost completely unknown, so any multi-compartment model is highly speculative

20. 2.4 Propagation of the AP in a passive and active axon Attenuation of 70% in 1mm, and very slow (0.2m/s) propagation Axon Electrodes V time Time + dt

21. 2.4 Propagation of the AP in a passive and active axon propagation Na+ Axon Electrodes V time Time + dt

22. 3 Synaptic conductances -Excitatory -Inhibitory Synaptic Current: Is = g(t) (Vm - Es) EPSC: AMPA (fast), NMDA (slow) IPSC: GABAA (fast) EPSC: g(t) is ~ an exponential

23. 4 Reduced models of neurons. Leaky Integrate and Fire. Models Stereotyped After Hyperpolarization Potential Models Stereotyped effects of incoming spikes Models synaptic channels g(t) A new spike occurs at time tnew if the threshold is reached: V is reset and integration begins again

24. 4 Reduced models of neurons. Leaky Integrate and Fire. Two spiking regimes: sub- and supra-threshold regimes Supra-threshold regime Sub-threshold regime

25. 4 Conductance-based I&F neuron Models Stereotyped After Hyperpolarization Potential Models stereotyped excitatory channels Models stereotyped inhibitory channels Few solutions were known for this model. But see recent developments by M. Richardson et al, Destexhe et al, and R. Moreno-Bote et al.

26. 4 Spike response neuron • Good approximation of I&F neuron model, but only with noisy inputs. • Spikes are generated randomly (Poisson) given the input u(t). Models Stereotyped After Hyperpolarization Potential Models stereotyped post-synaptic potentials f u

27. 5 Neuronal networks r(t) rate input τ dr/dt = -r + f( W r(t)+ W0 r0(t)) r0(t ) Exc + Inh pops.: E I

28. 5 Neuronal networks. Balanced regime Balanced regime: experimentally found that firing is low and irregular. Excitation in cortex is large. Then, excitation must be cancelled out by strong inhibition. Gerstein and Mandelbrot (1964), Van Vreeswijk and Sompolinsky (1996), Shadlen and Newsome (1998) only exc rE,out = rE,in rE,out balanced exc/inh rE,in Low variability regime High variability regime

29. 5 Neuronal networks. Balanced regime Itotal = (NEJErE - NIJIrI)m ~ Threshold N = 10000 JE = 0.2 mV r = 2-5Hz If (8000×0.2×2-2000×JI×5)×0.020=20mV -> JI = 0.22 mV If JI = 0.25 mV, then Itotal = 14 mV (No firing!) If JI = 0.19 mV, then Itotal = 26 mV (Saturation!) only exc rE,out = rE,in rE,out balanced exc/inh rE,in Problem: it requires fine-tuning of the network parameters (e.g., N, J…)

30. 5 Neuronal networks. Balanced regime rate input N, neurons K, connections Take the large N limit, with 1<<K<<N, and in particular

31. 5 Neuronal networks. Balanced regime N, neurons K, connections

32. Overview: • Passive properties of neurons (resting potential) • Action potential (generation and propagation). • Synaptic currents (AMPA, GABA, NMDA). • 4. Reduced models of neurons (LIF, QIF, LNP). • 5. Neuronal networks (balance and chaos).