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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