Neural codes and spiking models

Neuronal codes Spiking models: Hodgkin Huxley Model (small regeneration) Reduction of the HH-Model to two dimensions (general) FitzHugh-Nagumo Model Integrate and Fire Model Spike Response Model

Neuronal codes Spiking models: Hodgkin Huxley Model (small regeneration) Reduction of the HH-Model to two dimensions (general) FitzHugh-Nagumo Model Integrate and Fire Model Spike Response Model

Neuronal Codes – Action potentials as the elementary units voltage clamp from a brain cell of a fly

Neuronal Codes – Action potentials as the elementary units voltage clamp from a brain cell of a fly after band pass filtering

Neuronal Codes – Action potentials as the elementary units voltage clamp from a brain cell of a fly after band pass filtering generated electronically by a threshold discriminator circuit

Neuronal Codes – Probabilistic response and Bayes’ rule stimulus spike trains conditional probability:

Neuronal Codes – Probabilistic response and Bayes’ rule conditional probability ensembles of signals natural situation: joint probability: experimental situation: we choose s(t) prior distribution joint probability

Neuronal Codes – Probabilistic response and Bayes’ rule But: the brain “sees” only {t i } and must “say” something about s(t) But: there is no unique stimulus in correspondence with a particular spike train thus, some stimuli are more likely than others given a particular spike train experimental situation: response-conditional ensemble

Neuronal Codes – Probabilistic response and Bayes’ rule Bayes’ rule: what we see: what our brain “sees”:

Neuronal Codes – Probabilistic response and Bayes’ rule motion sensitive neuron H1 in the fly’s brain: average angular velocity of motion across the VF in a 200ms window spike count determined by the experimenter property of the neuron correlation

Neuronal Codes – Probabilistic response and Bayes’ rule spikes determine the probability of a stimulus from given spike train stimuli

Neuronal Codes – Probabilistic response and Bayes’ rule determine the probability of a stimulus from given spike train

Neuronal Codes – Probabilistic response and Bayes’ rule determine probability of a spike train from a given stimulus

Neuronal Codes – Probabilistic response and Bayes’ rule determine probability of a spike train from a given stimulus

Neuronal Codes – Probabilistic response and Bayes’ rule How do we measure this time dependent firing rate?

Neuronal Codes – Probabilistic response and Bayes’ rule Nice probabilistic stuff, but SO, WHAT?

Neuronal Codes – Probabilistic response and Bayes’ rule SO, WHAT? We can characterize the neuronal code in two ways: translating stimuli into spikestranslating spikes into stimuli Bayes’ rule: (traditional approach) -> If we can give a complete listing of either set of rules, than we can solve any translation problem thus, we can switch between these two points of view (how the brain “sees” it)

Neuronal Codes – Probabilistic response and Bayes’ rule We can switch between these two points of view. And why is that important? These two points of view may differ in their complexity!

Neuronal Codes – Probabilistic response and Bayes’ rule

average number of spikes depending on stimulus amplitude average stimulus depending on spike count

Neuronal Codes – Probabilistic response and Bayes’ rule average number of spikes depending on stimulus amplitude average stimulus depending on spike count non-linear relation almost perfectly linear relation That’s interesting, isn’t it?

Neuronal Codes – Probabilistic response and Bayes’ rule For a deeper discussion read, for instance, that nice book: Rieke, F. et al. (1996). Spikes: Exploring the neural code. MIT Press.

Neuronal codes Spiking models: Hodgkin Huxley Model (small regeneration) Reduction of the HH-Model to two dimensions (general) FitzHugh-Nagumo Model Integrate and Fire Model Spike Response Model

Hodgkin Huxley Model: with and charging current Ion channels

Hodgkin Huxley Model: (for the giant squid axon) with voltage dependent gating variables time constant asymptotic value

If u increases, m increases -> Na+ ions flow into the cell at high u, Na+ conductance shuts off because of h h reacts slower than m to the voltage increase K+ conductance, determined by n, slowly increases with increased u action potential

General reduction of the Hodgkin-Huxley Model stimulus 1) dynamics of m are fast 2) dynamics of h and n are similar

General Reduction of the Hodgkin-Huxley Model: 2 dimensional Neuron Models stimulus

FitzHugh-Nagumo Model u: membran potential w: recovery variable I: stimulus

FitzHugh-Nagumo Model nullclines

w u I(t)=I 0 FitzHugh-Nagumo Model nullclines stimulus

w u I(t)=0 For I=0: convergence to a stable fixed point FitzHugh-Nagumo Model nullclines

w u I(t)=I 0 limit cycle - unstable fixed point limit cycle FitzHugh-Nagumo Model stimulus FitzHugh-Nagumo Model nullclines

FitzHugh-Nagumo Model

The FitzHugh-Nagumo model – Absence of all-or-none spikes no well-defined firing threshold weak stimuli result in small trajectories (“subthreshold response”) strong stimuli result in large trajectories (“suprathreshold response”) BUT: it is only a quasi-threshold along the unstable middle branch of the V-nullcline (java applet)

The FitzHugh-Nagumo model – Excitation block and periodic spiking Increasing I shifts the V-nullcline upward -> periodic spiking as long as equilibrium is on the unstable middle branch -> Oscillations can be blocked (by excitation) when I increases further

The Fitzhugh-Nagumo model – Anodal break excitation Post-inhibitory (rebound) spiking: transient spike after hyperpolarization

The Fitzhugh-Nagumo model – Spike accommodation no spikes when slowly depolarized transient spikes at fast depolarization

Neuronal codes Spiking models: Hodgkin Huxley Model (small regeneration) Reduction of the HH-Model to two dimensions (general) FitzHugh-Nagumo Model Integrate and Fire Model Spike Response Model

i j Spike reception Spike emission Integrate and Fire model models two key aspects of neuronal excitability: passive integrating response for small inputs stereotype impulse, once the input exceeds a particular amplitude

i Spike reception: EPSP Fire+resetthreshold Spike emission reset I j Integrate and Fire model

i -spikes are events -threshold -spike/reset/refractoriness I(t) Time-dependent input Integrate and Fire model

If firing: I=0 u I>0 u resting t u repetitive t Integrate and Fire model (linear) u

linear non-linear If firing: Integrate and Fire model

Fire+reset non-linear threshold I=0 u I>0 u Quadratic I&F: Integrate and Fire model (non-linear)

I=0 u Integrate and Fire model (non-linear) critical voltage for spike initiation (by a short current pulse)

Fire+reset non-linear threshold I=0 u I>0 u Quadratic I&F: exponential I&F: Integrate and Fire model (non-linear)

Strict voltage threshold - by construction - spike threshold = reset condition There is no strict firing threshold - firing depends on input - exact reset condition of minor relevance Linear integrate-and-fire: Non-linear integrate-and-fire:

C glgl g Kv1 g Na I g Kv3 I(t) u Comparison: detailed vs non-linear I&F

Neuronal codes Spiking models: Hodgkin Huxley Model (small regeneration) Reduction of the HH-Model to two dimensions (general) FitzHugh-Nagumo Model Integrate and Fire Model Spike Response Model

Spike response model (for details see Gerstner and Kistler, 2002) = generalization of the I&F model SRM: parameters depend on the time since the last output spike integral over the past I&F: voltage dependent parameters differential equations allows to model refractoriness as a combination of three components: 1.reduced responsiveness after an output spike 2.increase in threshold after firing 3.hyperpolarizing spike after-potential

i j Spike reception: EPSP Spike emission: AP Firing: Spike emission Last spike of i All spikes, all neurons Spike response model (for details see Gerstner and Kistler, 2002) time course of the response to an incoming spike synaptic efficacy form of the AP and the after-potential

i j Spike response model (for details see Gerstner and Kistler, 2002) external driving current

Firing: threshold Spike response model – dynamic threshold

C glgl g Kv1 g Na I g Kv3 Comparison: detailed vs SRM I(t) detailed model Spike threshold model (SRM) <2ms 80% of spikes correct (+/-2ms)

References Rieke, F. et al. (1996). Spikes: Exploring the neural code. MIT Press. Izhikevich E. M. (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press. Fitzhugh R. (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1: Nagumo J. et al. (1962) An active pulse transmission line simulating nerve axon. Proc IRE. 50:2061–2070 Gerstner, W. and Kistler, W. M. (2002) Spiking Neuron Models. Cambridge University Press. online at: