”When spikes do matter: speed and plasticity” Thomas Trappenberg 1.Generation of spikes 2.Hodgkin-Huxley equation 3.Beyond HH (Wilson model) 4.Compartmental.

”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

Buracas, Zador, DeWeese, Albright, Neuron, 20:959-969 (1998)

Even without much information in spike trains

Spikes do matter !

Even if spikes matter

Rate models are well motivated !

Generation of a spike

Concentration gradient (Nernst equation)

Electrical force

Hodgkin-Huxley equations

Wilson model 1

Equilibrium potential Time constants

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)

Compartmental modelling

Neuron (and network) simulators

like NEURON and GENESIS

Cable equations + active channels

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

I=8 I=16I=12

Average current-frequency curve (activation,gain,transfer) - function

Poisson input spike trains

Fine-tuning of synaptic weights?

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)

Adapted from Abbott & Nelson, Nature Neuroscience Oct. 2000

Hebbian (asymmetric) learning 2

Hebbian (asymmetric) learning 3

Song & Abbott, Neurocomputing Oct. 2000

Variability control Gain control

Hebbian (asymmetric) learning 4

Van Rossum, Bi, & Turrigiano, J. Neuroscience, Dec. 2000

(Fokker-Planck equation)

Additive vs. Multiplicative rules ?

Hebbian (asymmetric) learning 5

Rate models 1

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)

Rate models 3

Fast processing

Panzeri, Rolls, Battaglia & Lavis, Network: Comput. Neural Syst. 12:423-440 (2001)

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