Basic elements of neuroelectronics -- membranes -- ion channels -- wiring. Elementary neuron models -- conductance based -- modelers alternatives

Transcription

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

2 Equivalent circuit model of a neuron

3 Closeup of a patch on the surface of a neuron

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 has an independent circuit path 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 nonlinearity in ion channels Voltage dependent transmitter dependent (synaptic) Ca dependent

11 The ion channel is a complex 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 Activation and inactivation dynamics We can rewrite: where

14 Activation and inactivation dynamics

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

16 The Hodgkin-Huxley equation

17 Activation and inactivation dynamics

18 Dynamics of a spike E K E Na

19 Ion channel stochasticity

20 A microscopic stochastic model for ion channel function approach to macroscopic description

21 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

22 The integrate-and-fire model 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.

23 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

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

25 Building circuits Eickholt lab, Kings College London

26 Synapses Signal is carried chemically across the synaptic cleft

27 Synaptic signalling

28 Synaptic signalling

29 Synaptic signalling

30 Synaptic signalling

31 Synaptic signalling

32 Synaptic signalling

33 Synaptic signalling

34 Synaptic signalling vesicle Neurotransmitter: glutamate AMPA receptor NMDA receptor Cation (Na) Ca

35 Post-synaptic conductances Requires pre- and post-synaptic depolarization

36 Connection strength w = npq

37 Long-term potentiation Wiki commons

38 Long-term depression Ronesi and Lovinger, J Physiol 2005

39 Empirical model Shouval,.., Cooper, Biological Cybernetics 2002

40 Hebbian plasticity Δ w ij = η x i x j Hebb, 1949

41 Post-synaptic conductances Δ w ij = η x i x j Requires pre- and post-synaptic depolarization Coincidence detection, Hebbian

42 Spike-timing dependent plasticity

43 Short-term synaptic plasticity Depression Facilitation

44 Modeling short-term synaptic plasticity Effective Recovered Inactive Tsodyks and Markram, 1997

45 Modeling short-term synaptic plasticity Tsodyks and Markram, 1997

46 Gap junctions Echevaria and Nathanson