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Basic elements of neuroelectronics
Elementary neuron models
-- conductance based -- modelers’ alternatives
Wiring neurons together
-- synapses -- long term plasticity -- short term plasticity
-- membranes -- ion channels -- wiring
Computing in carbon
Wires -- signal propagation -- processing in dendrites
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Equivalent circuit model of a neuron
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Closeup of a patch on the surface of a neuron
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Ohm’s law:
Capacitor: C = Q/V
Kirchhoff:
The passive membrane
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Energetics: qV ~ kBT
V ~ 25mV
Movement of ions through ion channels
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Na+, Ca2+ K+
Ions move down their concentration gradient
until opposed by electrostatic forces
Nernst:
The equilibrium potential
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Different ion channels have associated conductances.
A given conductance tends to move the membrane potential toward the equilibrium potential for that ion
V > E positive current will flow outward
V < E positive current will flow inward
ENa ~ 50mV ECa ~ 150mV EK ~ -80mV ECl ~ -60mV
depolarizing depolarizing hyperpolarizing shunting
V
Vrest
0
ENa
EK
more polarized
Each ion has an independent circuit path
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Several I-V curves in parallel:
New equivalent circuit:
Parallel paths for ions to cross membrane
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Neurons are excitable
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• Voltage dependent • transmitter dependent (synaptic) • Ca dependent
Excitability arises from nonlinearity in ion channels
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The ion channel is a complex molecular machine
Persistent conductance
K channel: open probability increases when depolarized
PK ~ n4
n is open probability 1 – n is closed probability
Transitions between states occur at voltage dependent rates
C O
O C
n describes a subunit
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Transient conductances
Gate acts as in previous case
PNa ~ m3h
Additional gate can block channel when open
m and h have opposite voltage dependences: depolarization increases m, activation hyperpolarization increases h, deinactivation
m is activation variable h is inactivation variable
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We can rewrite:
where
Activation and inactivation dynamics
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Activation and inactivation dynamics
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-
Ohm’s law: and Kirchhoff ’s law
Capacitative current
Ionic currents Externally applied current
Putting it together
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The Hodgkin-Huxley equation
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Activation and inactivation dynamics
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Dynamics of a spike
EK ENa
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Ion channel stochasticity
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A microscopic stochastic model for ion channel function
approach to macroscopic description
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Different from the continuous model: interdependence between inactivation and activation transitions to inactivation state 5 can occur only from 2,3 and 4 k1, k2, k3 are constant, not voltage dependent
Transient conductances
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The integrate-and-fire model
Like a passive membrane:
but with the additional rule that when V VT, a spike is fired
and V Vreset.
EL is the resting potential of the “cell”.
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• determine f from the linearized HH equations • fit a threshold • paste in the spike shape and AHP
Kernel f for subthreshold response replaces leaky integrator Kernel for spikes replaces “line”
The spike response model
Gerstner and Kistler
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Truccolo and Brown, Paninski, Pillow, Simoncelli
• general definitions for k and h • robust maximum likelihood fitting procedure
The generalized linear model
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Building circuits
Eickholt lab, Kings College London
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Signal is carried chemically across the synaptic cleft
Synapses
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Synaptic signalling
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Synaptic signalling
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Synaptic signalling
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Synaptic signalling
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Synaptic signalling
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Synaptic signalling
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Synaptic signalling
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Synaptic signalling
vesicle
Neurotransmitter: glutamate
AMPA receptor
NMDA receptor
Cation (Na)
Ca
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Requires pre- and post-synaptic depolarization
Post-synaptic conductances
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Connection strength
w = npq
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Long-term potentiation
Wiki commons
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Long-term depression
Ronesi and Lovinger, J Physiol 2005
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Empirical model
Shouval, .., Cooper, Biological Cybernetics 2002
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Hebbian plasticity
Hebb, 1949
Δ wij = η xi xj
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Requires pre- and post-synaptic depolarization
Coincidence detection, Hebbian
Post-synaptic conductances
Δ wij = η xi xj
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Spike-timing dependent plasticity
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Short-term synaptic plasticity
Depression Facilitation
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Modeling short-term synaptic plasticity
Tsodyks and Markram, 1997
Inactive Recovered Effective
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Modeling short-term synaptic plasticity
Tsodyks and Markram, 1997
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Gap junctions
Echevaria and Nathanson