Basic elements of neuroelectronics -- membranes -- ion channels -- wiring

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1 Computing in carbon Basic elements of neuroelectronics -- membranes -- ion channels -- wiring Elementary neuron models -- conductance based -- modelers alternatives Wires -- signal propagation -- processing in dendrites Wiring neurons together -- synapses -- long term plasticity -- short term plasticity

2 Equivalent circuit model

3 Membrane patch

4 The passive membrane Ohm s law: Capacitor: C = Q/V Kirchhoff:

5 Movement of ions through ion channels Energetics: qv ~ k B T V ~ 25mV

6 The equilibrium potential K + Na +, Ca 2+ Ions move down their concentration gradient until opposed by electrostatic forces Nernst:

7 Each ion type travels through independently Different ion channels have associated conductances. A given conductance tends to move the membrane potential toward the equilibrium potential for that ion E Na ~ 50mV E Ca ~ 150mV E K ~ -80mV E Cl ~ -60mV depolarizing depolarizing hyperpolarizing shunting V E Na 0 more polarized V > E positive current will flow outward V < E positive current will flow inward V rest E K

8 Parallel paths for ions to cross membrane Several I-V curves in parallel: New equivalent circuit:

9 Neurons are excitable

10 Excitability arises from ion channel nonlinearity Voltage dependent transmitter dependent (synaptic) Ca dependent

11 The ion channel is a cool molecular machine K channel: open probability increases when depolarized n describes a subunit n is open probability 1 n is closed probability Transitions between states occur at voltage dependent rates P K ~ n 4 C O O C Persistent conductance

12 Transient conductances Gate acts as in previous case Additional gate can block channel when open P Na ~ m 3 h m is activation variable h is inactivation variable m and h have opposite voltage dependences: depolarization increases m, activation hyperpolarization increases h, deinactivation

13 Dynamics of activation and inactivation We can rewrite: where

14 Dynamics of activation and inactivation

15 Putting it together Ohm s law: and Kirchhoff s law - Capacitative current Ionic currents Externally applied current

16 The Hodgkin-Huxley equation

17 Anatomy of a spike E K E Na Na ~ m 3 h K ~ m 3 h

18 Anatomy of a spike Runaway +ve feedback E K E Na Double whammy

19 Where to from here? Hodgkin-Huxley Biophysical realism Molecular considerations Geometry Simplified models Analytical tractability

20 Ion channel stochasticity

21 Microscopic models for ion channel fluctuations approach to macroscopic description

22 Transient conductances Different from the continuous model: interdependence between inactivation and activation transitions to inactivation state 5 can occur only from 2,3 and 4 k 1, k 2, k 3 are constant, not voltage dependent

23 The integrate-and-fire neuron Like a passive membrane: but with the additional rule that when V V T, a spike is fired and V V reset. E L is the resting potential of the cell.

24 Exponential integrate-and-fire neuron f(v) V reset V rest V th V max V f(v) = -V + exp([v-v th ]/D)

25 The theta neuron V spike V rest V th dq/dt = 1 cos q + (1+ cos q) I(t) Ermentrout and Kopell

26 The spike response model Kernel f for subthreshold response replaces leaky integrator Kernel for spikes replaces line determine f from the linearized HH equations fit a threshold paste in the spike shape and AHP Gerstner and Kistler

27 Two-dimensional models w Simple model: V = -av + bv 2 - cw W = -dw + ev V

28 The generalized linear model general definitions for k and h robust maximum likelihood fitting procedure Truccolo and Brown, Paninski, Pillow, Simoncelli

29 Dendritic computation

30 Dendritic computation Dendrites as computational elements: Passive contributions to computation Active contributions to computation Examples

31 Geometry matters Injecting current I 0 r V m = I m R m Current flows uniformly out through the cell: I m = I 0 /4pr 2 Input resistance is defined as R N = V m (t )/I 0 = R m /4pr 2

32 Linear cables r m and r i are the membrane and axial resistances, i.e. the resistances of a thin slice of the cylinder

33 Axial and membrane resistance c m r m r i For a length L of membrane cable: r i r i L r m r m / L c m c m L

34 The cable equation (1) x x+dx (2)

35 The cable equation (1) (2) (1) or where Time constant Space constant

36 General solution: filter and impulse response Exponential decay Diffusive spread

37 Voltage decays exponentially away from source Current injection at x=0, T 0

38 Properties of passive cables Electrotonic length

39 Electrotonic length Johnson and Wu

40 Properties of passive cables Electrotonic length Current can escape through additional pathways: speeds up decay

41 Voltage rise time Current can escape through additional pathways: speeds up decay Johnson and Wu

42 Properties of passive cables Electrotonic length Current can escape through additional pathways: speeds up decay Cable diameter affects input resistance

43 Properties of passive cables Electrotonic length Current can escape through additional pathways: speeds up decay Cable diameter affects input resistance Cable diameter affects transmission velocity

44 Step response: pulse travels Conduction velocity

45 Conduction velocity

46 Other factors Finite cables Active channels

47 Rall model Impedance matching: If a 3/2 = d 1 3/2 + d 2 3/2 can collapse to an equivalent cylinder with length given by electrotonic length

48 Active cables New cable equation for each dendritic compartment

49 Who ll be my Rall model, now that my Rall model is gone Genesis, NEURON

50 Passive computations London and Hausser, 2005

51 Enthusiastically recommended references Johnson and Wu, Foundations of Cellular Physiology, Chap 4 The classic textbook of biophysics and neurophysiology: lots of problems to work through. Good for HH, ion channels, cable theory. Koch, Biophysics of Computation Insightful compendium of ion channel contributions to neuronal computation Izhikevich, Dynamical Systems in Neuroscience An excellent primer on dynamical systems theory, applied to neuronal models Magee, Dendritic integration of excitatory synaptic input, Nature Reviews Neuroscience, 2000 Review of interesting issues in dendritic integration London and Hausser, Dendritic Computation, Annual Reviews in Neuroscience, 2005 Review of the possible computational space of dendritic processing

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