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