1. Persistent activity and oscillations in recurrent neural networks in the high-conductance regime Rubén Moreno-Bote with Romain Brette and Néstor Parga

2. Zoo of Membrane and Synaptic Time Scales Synaptic decay constants: Cortical neurons,  m  20 msPurkinje cells,  m  70 ms  GABA-A  10 ms  GABA-B  200 ms  AMPA  3 ms  NMDA  100 ms Membrane time constants:

3. A single pre-syn. neuron reduces  m = 53 ms = 25 ms = 9.1 ms = 5.6 ms Hausser and Clark, 1997

4.  m reduction in cortex  20 ms Passive membrane: Adding synaptic conductances: Background Activity produces a 5-fold increase of the membrane conductance (Pare et al, 1998; Destexhe and Pare, 1999)  4 ms Sensory Stimulation produces a further 2-fold increase (Borg-Graham et al, 1998; Hirsch et al, 1998; Anderson et al, 2000; Wehr and Zador, 2003; Tan et al, 2004)  2 ms

5. The High Conductance regime (HCR) -I study the high conductance regime when  m becomes shorter or comparable to the synaptic time constants - Main question: What are the properties of neurons and networks in this high conductance regime? - Why? Theoretical models have dealt only with the opposite limit  m >> synaptic time constants

6. Model. Conductance-based IF neuron The voltage of an conductance-based integrate-and-fire (IF) neuron with passive membrane time constant  m =C m /g L, and receiving Exc and Inh conductances g p E (t) and g p E (t) follows The neuron emits a spike when the voltage reaches a threshold value, Θ, after which the voltage is reset to a depolarized value H. time V threshold voltage

7. Model. Fluctuating conductances Excitatory and Inhibitory conductances (K = Exc, Inh) are modeled as OU processes with time scale  k followed by half-rectification: synaptic fluctuation The synaptic fluctuations have a 2D Normal distribution: Back + Stim Moreno-Bote and Parga, Physical Review Letters, 2004 and 2005.

8. Results. Rate of an IF neuron in the High Conductance Regime Instantaneous firing rate for an integrate-and-fire (IF) neuron receiving constant fluctuations Firing rate z, constant ensemble average Moreno-Bote and Parga, Physical Review Letters, 2005.

9. Results. A simpler formula for the rate  I approximate is large and positive in the subthreshold and high conductance regimes Firing rate u u min Moreno-Bote and Parga, Physical Review Letters, 2005.

10. Results. Firing rate as a function of the total conductance Moreno-Bote and Parga, Physical Review Letters, 2005.

11. Model. Conductance-based neural networks NENE NINI CECE CECE CICI CICI Sparse and random connectivity: Moreno-Bote et al, in preparation C p / N p << 1

12. Model. Conductance-based neural networks NENE NINI CECE CECE CICI CICI Sparse and random connectivity: The voltage of neuron i from population p=E,I, follows where  m =10ms is the passive membrane time constant. Moreno-Bote et al, in preparation C p / N p << 1

13. Model. Conductance-based neural networks NENE NINI CECE CECE CICI CICI Sparse and random connectivity: The voltage of neuron i from population p=E,I, follows where  m =10ms is the passive membrane time constant. Synaptic conductances increase by a fixed amount per incoming spike and then decay exponentially with time constant  s,Exc =5ms or  s,Inh =10ms: Extenal conductances follow Moreno-Bote et al, in preparation C p / N p << 1

14. Results. Mean-field theory for neural networks in the HCR (1) If a neuron in population p=E,I receives synaptic conductances g p E (t) and g p I (t) at some time t, the firing rate of the population at that time is: with The mean-field consist of (1)Replacing the spikes from population p by its population firing rate, and (2) Averaging the synaptic conductances over the population. (2) The mean conductance over the population exponentially filters the population firing rates: Reduction from 2N eqs. to 2 eqs.!!

15. Results. Spontaneous activity and transients Moreno-Bote et al, in preparation stimulus intensity

16. Results. Spontaneous and persistent activity in recurrent networks Moreno-Bote et al, in preparation stimulus intensity NENE NENE

17. Results. Oscillations in a purely inhibitory recurrent network with delays in the HCR Moreno-Bote et al, in preparation

18. … in conclusion 1.We have calculated the firing rate of an IF neuron in the high conductance regime 2.This simple equation allows for a mean-field description of conductance-based neuronal network in the high-conductance regime. 3.The mean-field quantitatively describes the spontaneous and persistent activity states of neuronal networks that have memory-like states. The mean-field also predicts under which conditions the memory-like states become stable or unstable. 4.The mean-field predicts the oscillation frequency of conductance-based neuronal networks with synaptic delays. 5.A similar mean-field theory can be developed for current-based neuronal networks.