When spikes do matter:

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When spikes do matter
speed and plasticity
Thomas Trappenberg

• Generation of spikes
• Hodgkin-Huxley equation
• Beyond HH (Wilson model)
• Compartmental model
• Integrate-and-fire model
• Hebbian (asymmetric) learning
• Population rate models

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Buracas, Zador, DeWeese, Albright, Neuron,
20959-969 (1998)
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Even without much information in spike
trains Spikes do matter ! Even if spikes
matter Rate models are well motivated !
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Generation of a spike
Concentration gradient (Nernst equation)
Electrical force
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Hodgkin-Huxley equations
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Wilson model 1
Equilibrium potential
Time constants
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Wilson model 2
Na leakage and voltage dependent channel K
voltage dependent channel with slow dynamic Ca2
voltage dependent channel with slow dynamics K
dynamic voltage dependent channel (Ca2 mediated)
Hugh R. Wilson Simplified Dynamics of Human and
Mammalian Neocortical Neurons J. Theoretical
Biology 200 375-388 (1999)
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Compartmental modelling
Neuron (and network) simulators like NEURON and
GENESIS
Cable equations active channels
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Integrate-and-fire neuron
(see also spike-response model)
1. Sub-threshold leaky-integrator dynamic 2.
Summation of PSPs from synaptic input 3. Firing
threshold (spike generation) 4. Reset of
membrane potential
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Average current-frequency curve
(activation,gain,transfer) - function
I8
I16
I12
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Poisson input spike trains
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Hebbian (asymmetric) learning 1
The organization of behavior (1949) When an axon
of a cell A is near enough to excite cell B or
repeatedly or persistently takes part in firing
it, some growth or metabolic change takes place
in both cells such that A's efficiency, as one of
the cells firing B, is increased.
Donald Hebb (1904-1985)
G.-q. Bi and M.-m. Poo, J. of Neuroscience
1810464-10472 (1998)
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Hebbian (asymmetric) learning 2
Adapted from Abbott Nelson, Nature
Neuroscience Oct. 2000
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Hebbian (asymmetric) learning 3
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Hebbian (asymmetric) learning 4
Variability control
Gain control
Song Abbott, Neurocomputing Oct. 2000
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Hebbian (asymmetric) learning 5
Additive vs. Multiplicative rules ?
Van Rossum, Bi, Turrigiano, J. Neuroscience,
Dec. 2000
(Fokker-Planck equation)
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Rate models 1
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Rate models 2
1. 2. 3. 4.

• Population of similar neurons (e.g. same
  input, same time constant, )
• Independent (e.g. no locking, synchronization,
  no sigma-pi,
• Write as integral equation (e.g. use spike
  response model see W. Gerstner)
• Mean field theory (e.g. averaging)
• Adiabatic limit (e.g. slow changes)

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Rate models 3
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Fast processing
Panzeri, Rolls, Battaglia Lavis, Network
Comput. Neural Syst. 12423-440 (2001)
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Conclusions

• Rate models are now well motivated
• Spike models are now well developed
• Hebbian plasticity is now better explored
• Spikes are important for rapid and robust
  information processing