single neuron modelsmathneuronet.org.uk/training/dendrites2010.pdf · complex spatial structure...
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Spatially extended models of single neurons:
the role of dendritesYulia Timofeeva (Warwick, UK)
Mathematical Neuroscience Training Workshop 2010
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• The conductance-based membrane model
• Spatially extended models
Overview
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Complex spatial structure
Action potentials (short electrical spikes)
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The contacts of the axon to target neurons are either located on the dendritic tree or directly on the soma, and are known as synapses
Ionic gates are embedded in the cell membrane and control the passage of ions
Differences in the ionic concentrations of the intra/extracellular fluids create a potential difference across the cell
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Action potential
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Experimental setup in vitro
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Single-compartment models
The Nernst potential
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Diffusion of K+ ions down the concentration gradient through the membrane (a) creates an electric potential force directed at the opposite direction (b) until the diffusion and electrical forces counter each other (c) resulting in the Nernst equilibrium potential for K+
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Passive redistribution and active transport support the concentration asymmetry
Nernst potentials = Reversal potentials
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The membrane model
Ohm’s law:
- conductance
1. the phospholipid bilayer, which is analogous to a capacitor in that it accumulates ionic charge
2. the ionic permeabilities of the membrane, which are analogous to resistors
3. the electrochemical driving forces, which are analogous to batteries driving the ionic currents
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the current flow through a single K+ channel
- the conductance of the K+ channel
- the K+ driving force across the membrane
the capacitive current across the membrane
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The main equation of the membrane model
The Hodgkin-Huxley model
1949, Plymouth
Alan HodgkinAndrew HuxleyThey established experimentally the voltage dependence of ion conductances in the electrically excitable membrane of the squid giant axon
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A gated ionic channel
the transition from state C to state Othe transition from state O to state C
- the fraction of open channels
rate of change = inflow rate - outflow rate
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Voltage-gated channel
and
The great insight of Hodgkin and Huxley was to realise that depends upon four activation gates:
depends upon three activation gates and one inactivation gate:
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The Hodgkin-Huxley model
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Basic functions of neurons
1. Generate intrinsic activity
2. Receive synaptic inputs
3. Integrate signals
4. Encode output patterns
5. Distribute synaptic outputs
From G. M. Shepherd
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From Mainen and Sejnowski, 1996
Distinct firing patternsLayer 3 spiny stellate
Layer 4 spiny stellate
Layer 3 pyramidal
Layer 5 pyramidal
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Spatially extended models
Isopotential somaNon-isopotential structure
Wilfrid Rall
Passive membraneActive membrane
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Linear cable theory
Using current balance
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space constant
membrane time constant
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Infinite cable and constant current at
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Infinite cable and delta-pulse stimulus
Solution – Green’s function
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Infinite cable and an arbitrary stimulus
convolution of initial data
convolution of stimulus
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The Rall model
Equivalent cable
3/2 law
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Node
Terminal
continuity of potentials & conservation of current
‘Sum-over-paths’ approach (L.F. Abbott,1992)
Trips
An arbitrary tree
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Coefficients Factor of segment
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A rat CA1 hippocampal pyramidal cell visualised with differential interference contrast optics using infrared illumination
Dual simultaneous whole-cell patch-clamp recordings
Dendritic & somatic recordings with respect to rest (at about -70 mV)
Quasi-active dendrites. Motivation
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nonlinear current
(with a single gating variable)
Quasi-active membrane
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On each branch
Using Laplace transform For example, for infinite cable
Recovers expected result as Passive system
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Seek a solution in terms of
Sum-over-trips (quasi-active membrane)
+1 (-1)
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Resonance associated with current
Model of nonlinear current ( Magee (1998) Journal of Neuroscience 18 )
- a single gating variable
Dashed line: Magee’s currentSolid line: ‘LRC’ circuit
Application
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Natural frequency
Resonant frequency
Two semi-infinite branches
Idealised geometry
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Two semi-infinite resonant branches
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Three semi-infinite resonant branches
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S Coombes, YT, C-M Svensson, G J Lord, K Josic, S J Cox and C M Colbert Biological Cybernetics (2007) Vol 97, pp. 137-149
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Andreas Herzog http://iesk.et.uni-magdeburg.de/~herzog/
Learning and memory, logical computations, pattern matching, amplification of distal synaptic inputs, temporal filtering
Action potentials in spine apparatus seen using Calcium dyes + confocal microscopy
Experimental observations of travelling waves in distal dendritic trees For a perspective see Segev & Rall ‘Excitable dendrites and spines: earlier theoretical
insights elucidate recent direct observations ’Trends in Neuroscience 21(11), 1998
Ramon y Cajal, 1896
Spiny dendrites
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Cable eqn. + currents from spines
Baer & Rinzel J.Neurophysiol.,1991
HH dynamics at spines
Baer&Rinzel model
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Continuum model Discrete model
Variations in spine distribution
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Motivation – to develop a simplified model, still
biophysically realistic but computationally cheap!
Spine-head (active)
Dendritic cable (passive)
- action potential from active spines
Integrate-and-Fire
The Spike-Diffuse-Spike model
Coombes & Bressloff SIAM J Appl Math 2000, PRL 2003
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Spine-head dynamics (IF)
Firing times:
- refractory time
Reset: whenever
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SDS model
BR model
YT, Lord & Coombes 2006 J.Comput.Neurosci. 21
Self-consistent speed
Spines:Firing events:Speed:
LP
Saltatory wave
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Irregular distribution of spines
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Compartmental models
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Simulation software tools
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I. Segev, J. Rinzel and G. M. Shepherd, The Theoretical Foundation of Dendritic Function: Selected Papers of Wilfrid Rall with Commentaries, MIT Press, Cambridge, MA, 1995
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H.C. Tuckwell, Introduction to theoretical neurobiology. Volume 1: Linear cable theory and dendritic structure, Cambridge Studies in Mathematical Biology, 1988
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G. Stuart, N. Spruston, M. Hausser, Dendrites, Oxford University Press, 2008