Convergence and stability in networks with spiking neurons Stan Gielen Dept. of Biophysics Magteld Zeitler Daniele Marinazzo

Overview What’s the fun about synchronization ? Neuron models Phase resetting by external input Synchronization of two neural oscillators What happens when multiple oscillators are coupled ? Feedback between clusters of neurons Stable propagation of synchronized spiking in neural networks Current problems

The neural code Firing rate Recruitment Synchronous firing Neuronal assembly Neuronal assemblies are flexible

Why flexible synchronization ? Stimulus driven; bottom-up process From Fries et al. Nat Rev Neurosci.

Synchronization of firing related to attention Riehle et al. Science, 1999 Evidence for Top-Down processes on coherent firing

Coherence between sensori-motor cortex MEG and muscle EMG Schoffelen, Oosterveld & Fries, Science in press Before and after a visual warning signal for the “go” signal to start a movement

Functional role of synchronization Schoffelen, Oosterveld & Fries, Science in press

Questions regarding initiation/disappearence of temporal coding Bottom-up and/or top-down mechanisms for initiation of neuronal synchronization ? Stability of oscillations of neuronal activity functional role of synchronized neuronal oscillations

Overview What’s the fun about synchronization ? Neuron models Phase resetting by external input Synchronization of two neural oscillators What happens when multiple oscillators are coupled ? Feedback between clusters of neurons Stable propagation of synchronized spiking in neural networks Current problems

Leaky-Integrate and Fire neuron For constant input I Small inputLarge input

Conductance-based Leaky- Integrate and Fire neuron Membrane conductance is a function of total input, and so is the time- constant. With increasing synaptic input, the neuron changes from an integrator to a co-incidence detector.

Synaptic processes

Conductance-based Leaky- Integrate and Fire neuron Membrane conductance is a function of total input, and so is the time- constant. With increasing synaptic input, the neuron changes from an integrator to a co-incidence detector.

Conductance-based Leaky- Integrate and Fire neuron With increasing synaptic input, the neuron changes from an integrator to a co-incidence detector. τ = 40 ms τ = 2 ms

Hodgkin-Huxley neuron

V mV 0 mV V mV 0 mV ICIC I Na Membrane voltage equation -C m dV/dt = g max, Na m 3 h(V-V na ) + g max, K n 4 (V-V K ) + g leak (V-V leak ) K

V (mV) mm mm Open Closed mm mm m Probability: State: (1-m) Channel Open Probability: mm mm Gating kinetics m.m.m.h=m 3 h

Actionpotential

Simplification of Hodgkin-Huxley Fast variables membrane potential V activation rate for Na + m Slow variables activation rate for K + n inactivation rate for Na + h -C dV/dt = g Na m 3 h(V-E na )+g K n 4 (V-E K )+g L (V-E L ) + I dm/dt = α m (1-m)-β m m dh/dt = α h (1-h)-β h h dn/dt = α n (1-n)-β n n Morris-Lecar model

Phase diagram for the Morris-Lecar model

Linearisation around singular point :

Phase diagram

Phase diagram of the Morris- Lecar model

Overview What’s the fun about synchronization ? Neuron models Phase resetting by external input Synchronization of two neural oscillators What happens when multiple oscillators are coupled ? Feedback between clusters of neurons Stable propagation of synchronized spiking in neural networks Current problems

Neuronal synchronization due to external input T ΔT Δ(θ)= ΔT/T Synaptic input

Neuronal synchronization T ΔT Δ(θ)= ΔT/T Phase shift as a function of the relative phase of the external input. Phase advance Hyperpolarizing stimulus Depolarizing stimulus

Neuronal synchronization T ΔT Δ(θ)= ΔT/T Suppose: T = 95 ms external trigger: every 76 ms Synchronization when ΔT/T=(95-76)/95=0.2 external trigger at time 0.7x95 ms = 66.5 ms

Example T=95 ms P=76 ms = T(95 ms) - Δ(θ) For strong excitatory coupling, 1:1 synchronization is not unusual. For weaker coupling we may find other rhythms, like 1:2, 2:3, etc.

Neuronal synchronization T ΔT Δ(θ)= ΔT/T Suppose: T = 95 ms external trigger: every 76 ms Synchronization when ΔT/T=(95-76)/95=0.2 external trigger at time 0.7x95 ms = 66.5 ms Stable Unstable

Convergence to a fixed-point Θ * requires Substitution of and expansion near gives Convergence requires and constraint gives T P

Overview What’s the fun about synchronization ? Neuron models Phase resetting by external input Synchronization of two neural oscillators What happens when multiple oscillators are coupled ? Feedback between clusters of neurons Stable propagation of synchronized spiking in neural networks Current problems

Excitatory/inhibitory interactions excitation-excitation inhibition-inhibition excitation-inhibition Behavior depends on synaptic strength ε and size of delay Δt

Excitatory interactions excitation-excitation Mirollo and Strogatz (1990) proved in a rigorous way that excitatory coupling without delays always leads to in-phase synchronization.

Stability for two excitatory neurons with delayed coupling Return map For two neurons with excitatory coupling strength = 0.1 and time delay = 0.2 Ernst et al. PRL 74, 1995 if t k is time when oscillator A fires

Summary for excitatory coupling between two neurons In-phase behavior for excitatory coupling without time delays tight coupling with a phase-delay for time delays with excitatory coupling.

Inhibitory interactions excitation-excitation

Inhibitory coupling for two identical leaky-integrate-and-fire neurons Out-of-phase stableIn-phase stable Lewis&Rinzel, J. Comp. Neurosci, 2003

Phase-shift function for neuronal synchronization T ΔT Δ(θ)= ΔT/T Phase shift as a function of the relative phase of the external input. Phase advance Hyperpolarizing stimulus Depolarizing stimulus

Phase-shift function for inhibitory coupling for stable attractor Increasing constant input to the LIF- neurons I=1.2 I=1.4 I=1.6

Bifurcation diagram for two identical LIF-neurons with inhibitory coupling

Time constant for inhibitory synaps

Summary for inhibitory coupling Stable pattern corresponds to out-of-phase synchrony when the time constant of the inhibitory post synaptic potential is short relative to spike interval in-phase when the time constant of the inhibitory post synaptic potential is long relative to spike interval

Inhibitory coupling with time delays

Stability for two inhibitory neurons with delayed coupling Return map For two neurons with excitatory coupling strength = 0.1 and time delay = 0.2 Ernst et al. PRL 74, 1995

Stability for two excitatory neurons with delayed coupling Return map For two neurons with excitatory coupling strength = 0.1 and time delay = 0.2 Ernst et al. PRL 74, 1995 if t k is time when oscillator A fires

Summary about two-neuron coupling with delays Excitation leads to out-of-phase behavior Inhibition leads to in-phase behavior

Overview What’s the fun about synchronization ? Neuron models Phase resetting by external input Synchronization of two neural oscillators What happens when multiple oscillators are coupled ? Feedback between clusters of neurons Stable propagation of synchronized spiking in neural networks Current problems

A network of oscillators with excitatory coupling

Winfree model of coupled oscillators P(Θ j ) is effect of j-th oscillator on oscillator I (e.g. P(Θ j ) =1+cos(Θ j ) R(Θ i ) is sensitivity function corresponding to contribution of oscillator to mean field. N-oscillators with natural frequency ω i Ariaratnam & Strogatz, PRL 86, 2001

Averaged frequency ρ i as a function of ω i lockingpartial locking incoherence partial death slowest oscillators stop Frequency range of oscillators Ariaratnam & Strogatz, PRL 86, 2001

Phase diagram assuming uniform distribution of natural frequencies

Summary A network of spontaneous oscillators with different natural frequencies can give –locking –partial locking –incoherence –partial death depending on strength of excitatory coupling and on distribution of natural frequencies. Excitatory coupling can cause synchrony and chaos !

Role of excitation and inhibition in neuronal synchronization in networks with excitatory and inhibitory neurons Borgers & Kopell Neural Computation 15, 2003

Role of excitation and inhibition in neuronal synchronization in networks with excitatory and inhibitory neurons g EE g IE g II g EI

Mutual synchronization All-to-all connectivitySparse connectivity I E =0.1; I I =0; g EI =g IE =0.25; g EE =g II =0; τ E =2 ms; τ I =10 ms g EE g IE g II g EI

Main message Synchronous rhythmic firing results from E-cells driving the I-cells I-cells synchronizing the E-cells Synchronization is obtained for Continuous drive to E-neurons Relatively strong E  I connections Short time decay of inhibitory post synaptic potentials

Simple (Theta) neuron model Neuronal state represented as phase on the unit-circle with input I (in radians) and membrane time-constant τ. When I<0: two fixed points : unstable stable I<0I=0 I>0 Saddle- node bifurcation : spiking neuron

unstable stable I<0 I=0 I>0 Spiking neuron Saddle- node bifurcation

Time interval between spikes -π π π =

Sufficient conditions for synchronized firing E=>I synapses too weakExternal input to I-cells I=>E synapses too weak Rhythm restored by adding I=>I synapses Conditions for synchrony E-cells receive external input above threshold I-cells spike only in response to E-cells; Relatively strong E  I connections I=>E synapses are short and strong such that I-cells synchronize E-cells

Overview What’s the fun about synchronization ? Neuron models Phase resetting by external input Synchronization of two neural oscillators What happens when multiple oscillators are coupled ? Feedback between clusters of neurons Stable propagation of synchronized spiking in neural networks Current problems

Possible role of feedback Data from electric fish Correlated input Uncorrelated input Doiron et al. Science, 2004

Possible role of feedback feedback noise Doiron et al. Science, 2003 Doiron et al. PRL, 93, 2004

Feedback to retrieve correlated input datamodel Experimental data

Linear response analysis gives Power in range 2-22 Hz Hz

What happens to our analytical formula when FB comes from a LIF neuron?

τ Delay = 18 ms Feedback gain = -1.2 τ LIF = 6 ms (longer time constant, same result)

Paradox with between results by Kopell (2004) and Doiron (2003, 2004) ? Börgers and Kopell (2003): spontaneous synchronized periodic firing in networks with excitatory and inhibitory neurons Doiron et al: Feedback serves to detect common input: no common input  no synchronized firing.

Short time constant for inhibitory neuron: τ LIF = 2 ms LIF Synchronized firing even without correlated input C=0

LIF Uncorrelated input Fully correlated input For small time constant of LIF-neuron, network starts spontaneous oscillations of synchronized firing

Sufficient conditions for synchronized firing in the Kopell model E=>I synapses too weakExternal input to I-cells I=>E synapses too weak Rhythm restored by adding I=>I synapses Conditions for synchrony E-cells receive external input above threshold I-cells spike only in response to E-cells; Relatively strong E  I connections I=>E synapses are short and strong such that I-cells synchronize E-cells

No feedback; input at 50, 60 or 90 HzWith feedback; input at 50, 60 or 90 Hz LIF LIF with small τ

Time constant of inhibitory neuron is crucial ! Short time constant: neuron is co-incidence detector Börgers and Kopell (2003): spontaneous synchronized periodic firing in networks with excitatory and inhibitory neurons Long time constant: Doiron et al: Feedback serves to detect common input: no common input  no synchronized firing.

Overview What’s the fun about synchronization ? Neuron models Phase resetting by external input Synchronization of two neural oscillators What happens when multiple oscillators are coupled ? Feedback between clusters of neurons Stable propagation of synchronized spiking in neural networks Current problems

Propagation of synchronous activity Diesman et al., Nature, 1999 Leaky integrate-and-fire neuron: 20,000 synpases; 88% excitatory and 12 % inhibitory Poisson-like output statistics ‘Synfire chain’

Propagation of synchronous activity Activity in 10 groups of 100 neurons each. Under what conditions preservation extinction of synchronous firing ?

Critical parameters Number of pulses in volley (‘activity’) temporal dispersion σ background activity integration time constant for neuron ( τ = 10 ms)

activity a dispersion σ Spike probability versus input spike number as function of σ Temporal accuracy versus σ in for various input spike numbers

Output less precise Output more precise

State-space analysis Model parameters: # 100 neurons Stable attractor Transmission function for pulse- packet for group of 100 neurons. Evolution of synchronous spike volley

State-space analysis Model parameters: # 100 neurons Attractor Saddle point

Dependence on size of neuron groups a minimum of 90 neurons are necessary to preserve synchrony fixed point depends on a, σ and w. N=80N=90N=100N=110 a-isocline σ -isocline

Summary Stable modes of coincidence firing. Attractor states depend on number of neurons involved, firing rate, dispersion and time constant of neurons.

Further questions What happens for correlated input from multiple groups of neurons ? What is the effect of (un)correlated excitation and inhibition ? What is the effect of lateral interactions ? What is the effect of feedback ?

Summary Excitatory coupling leads to chaos; inhibition lead to synchronized firing. Synchronization can easily be obtained by networks of coupled excitatory and inhibitory neurons. In that case the frequency of oscillations depends on the neuronal dynamics and delays, not on input characteristics There is no good model yet, which explains the role of input driven (bottom-up) versus top-down processes in the initiation of synchronized oscillatory activity. The role and relative contribution of feedforward (stochastic resonance) and feedback in neuronal synchronization is yet unknown.