”when spikes do matter: speed and plasticity” thomas trappenberg 1.generation of spikes...
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”When spikes do matter: speed and plasticity”
Thomas Trappenberg
1. Generation of spikes
2. Hodgkin-Huxley equation
3. Beyond HH (Wilson model)
4. Compartmental model
5. Integrate-and-fire model
6. Hebbian (asymmetric) learning
7. Population rate models
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Buracas, Zador, DeWeese, Albright, Neuron, 20:959-969 (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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I=8 I=16I=12
Average current-frequency curve (activation,gain,transfer) - function
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Poisson input spike trains
Fine-tuning of synaptic weights?
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Donald Hebb (1904-1985)
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.”
Hebbian (asymmetric) learning 1
G.-q. Bi and M.-m. Poo, J. of Neuroscience 18:10464-10472 (1998)
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Adapted from Abbott & Nelson, Nature Neuroscience Oct. 2000
Hebbian (asymmetric) learning 2
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Hebbian (asymmetric) learning 3
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Song & Abbott, Neurocomputing Oct. 2000
Variability control Gain control
Hebbian (asymmetric) learning 4
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Van Rossum, Bi, & Turrigiano, J. Neuroscience, Dec. 2000
(Fokker-Planck equation)
Additive vs. Multiplicative rules ?
Hebbian (asymmetric) learning 5
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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. 12:423-440 (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