Basic Issues in (Computational ) Neuroscience

1. Basic Issues in (Computational ) Neuroscience • What is the logics of the brain ? • How similar is the brain with conventional computers ? • How reliable are neurons, synapses,.... ? • How the brain is so smart although using unreliable components ? • How specific is electrical and chemical signalling ? • What is the elementary computational element ? A population of neurons or single neurons or part of a neuron ? • -......

2. Neuronal firing is not precise. Why then actions are reliable ? Why we do not make many mistakes ?

3. How compartimentalized are neurons ?How specific is signalling ?

4. In order to answer to these questions, we have • To analyse basic properties of neurons and synapses. • To understand their dynamics • To understand the origin of noise in neurons

5. Cell membranes Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm2 Ion channels  conductance

6. Membrane potential Fixed potential  concentration gradient Concentration difference  Potential difference Concentration difference maintained by ion pumps, which are transmembrane proteins

7. Nernst potential No ionic flow considered. The concentration of the ionic species is n1 and n2 inside and outside. At equilibrium we have: b = 1/kBT (q = proton charge, z = ionic charge in units of q)  No flow at this potential difference Called Nernst potential or reversal potential for that ion. This equation is a consequence of thermodynamics and of the Coupling between the electric flow and diffusion

8. Reversal potentials Note: VT = kBT/q = (for chemists) RT/F ~ 25 mv Some reversal potentials: K: -70 to -90 mV Na: +50 mV Cl: -60 to -65 mV it changes during development Ca: 150 mV Rest potential: ~ -65 mV ~2.5 VT How the membrane potential can reach values below -100 mV ?

9. More general flow equations Current through membrane: r = ion density Diffusive part: D = diffusion constant Drift in field: v = velocity m = mobility, F = force z = valence, e = proton charge, V = electrostatic potential Total current: Nernst-Planck equation

10. Nernst-Planck equation Can also be written using Einstein relation or where is the electrochemical potential

11. Exercise:Derive the Nernst equation Consider a membrane separating two solutions where the electrochemical potential is m1and m2. At the steady state ( or at equlibrium ) we have m1 = m2 And …………..

12. Constant Field (GHK) EquationGHK stands for Goldman Hodgkin Katz Nernst-Planck equation: Use integrating factor  Integrate from x0 to x1:

13. The integral can be computed assuming a constant field in the membrane V = membrane potential, d = membrane thickness can integrate denominator x1 = 0, x2 = d Result: vanishes at reversal potential, by definition. OSS: The relation between the flow ( or current ) and voltage is not linear

14. The nonlinear GHK equation can be linearizedaround the resting voltage Vr Using the Nerst equation i.e.,  Now expand in V-Vr we obtain a linear relation between J and V

15. Exercise:Plot the GHK equation for Na and K ions

16. Effective circuit model for cell membrane (“point model”: ignores spatial structure) (C, gi, Iext all per unit area) gi can depend on V, Ca concentration, synaptic transmitter binding, …

17. Ohmic model One gi = g = const or membrane time const Start at rest: V= V0 at t=0 Final state: Solution:

18. Exercises and Questions:When the opening of a single channel could initiate a spike ?What the voltage does when you inject a step of current ?What is the role of the membrane time constant ?

19. Voltage-dependent channels

20. Full Hodgkin-Huxley model What can we say from the voltage dependence of t and n ?

21. Questions:Whis hypotheses underlie HH equations ?Which properties of ionic channels the HH equations predicted properly ?

22. a and b are rate constants of transition. t is a relaxation time constant and n is the asymptotic value

23. Hodgkin-Huxley K channelActivation (n) variable P = n4 (solid: exponential model for both a and b Dashed: HH fit)

24. Hodgkin Huxley Na channel Activation (m) and inactivation (h) variables HH fits: m is fast (~.5 ms) h,n are slow (~5 ms)

25. Mechanisms of Spike generation • Current flows in, raises V • m increases (h slower to react)  gNa increases • more Na current flows in  … • V rises rapidly toward VNa Then h starts to decrease  gNa shrinks • V falls, aided by n opening for K current Overshoot, recovery Threshold effect What is the origin of the overshoot ?

26. Role of m,h and n inspike generation

27. Regular firing, rate vs Iext What is the origin of the threshold ? Is the firing always regular ?

28. Step increase in current:subtreshold and superthreshold behaviours What is the origin of the ringing behaviour ?

29. Noisy input current, refractoriness What is the origin of refractoriness ? Try to explain it in terms of m,n and h

30. Inventory of Ionic currentsDifferent properties correspond to distinct molecular mechanismsTry to relate macroscopicproperties to specific behavioursof m,n and h They are an essential feature of brain complexity

31. Persistent (noninactivating) Na channel NaV1.1 and NaV1.6 No h! Increases neuronal excitability

32. K channels: “A currents”Kv1.4 Kv3.3 (same form as HH Na channel) fast slow-inactivating current 2 kinds of each

33. Effect of A currents th ~ 10-20 ms Opposite direction from Na current: hyperpolarizes membrane Slows spike initiation: have to wait for IA to inactivate:

34. Type I and Type II neurons Type I: arbitrarily slow rate possible Type II: minimum firing rate >0

35. Ca2+ -dependent K conductancesBK channels (1): IC (persistent) Activation is [Ca2+]-dependent [Ca2+] = 0.1, 0,2, 0.5, 1.0, 2.0, 5.0 mmol/l Contributes to repolarization after spikes

36. Ca2+ -dependent K conductances (2): IAHP After-hyperpolarization current Slow, no voltage dependence! Ca2+ enters (through other channels) during action potentials Each spike  bigger a, bigger m  slows down spiking

37. Ca2+ currents (1): low-threshold ITT-type Ca Channels (ohmic approximation here, but see later) Closed at rest because h nearly 0 (channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0(channel is “not activated”) Consequences: (1) “Post-inhibitory rebound”; fires “Ca spike” on release from hyperpolarization (2) Ca spikes can lead to Na spikes

38. Ca2+ currents (2): high-threshold IL in ohmic approximation Persistent: Lets in some Ca2+ with each action potential This activates Ca-dependent K current Ca2+ dynamics: