In the Mean-driven limit, the population firing rates M S and M C obey

1. I I E E V1 Complex Simple LGN Excitatory Inhibitory Orientation Selectivity in Visual Cortex by Fluctuation-Controlled Criticality Louis Tao1, David Cai2, D. McLaughlin2,3, M. Shelley2,3 & R. Shapley3 1Department of Mathematical Sciences, NJIT; 2Courant Institute of Mathematical Sciences, NYU; 3Center for Neural Science, NYU Orientation Selectivity and Fluctuation-Controlled Criticality 1. Neurons in Ring Model are contrast invariant: Cortical gain can be adjusted near criticality 2. The Sparsified Egalitarian Model: neuronal selectivity is contrast invariant (not shown) Theory can be used to avoid Bistable Network models Bifurcations and a Fluctuation-Controlled Critical State Abstract: We examine how intrinsic synaptic fluctuations modify the effects of strong recurrent network amplification to produce orientation selectivity in a large-scale neuronal network model of the macaque primary visual cortex (V1). Strong cortical amplification can lead to network instabilities, even in the presence of strong cortical inhibition. In this poster, we show that large intrinsic fluctuations in the cortico-cortical conductances can stabilize the network, allowing strong cortical gain and the emergence of orientation selective neurons. By increasing the strength of the synaptic fluctuations, through sparsifying the network connectivity, we identify a transition between two types of dynamics, mean- and fluctuation-driven. In a network with strong recurrent excitation, this fluctuation-controlled transition is signified by a near hysteresis behavior and a rapid rise of network firing rates as the synaptic drive or stimulus input is increased. We discuss the connection between this fluctuation-controlled bifurcation and orientation selectivity in network models of V1. In a network where each neuron is coupled to N other neurons, and where the bifurcation changes as N is decreased and the intrinsic synaptic fluctuations are increased. Right. Tuning and Contrast Invariance of cells in a Ring Model: upper panels Exc. Simple (rates at 4 contrasts, membrane potential, conductances: green, LGN; red, cortical exc.; blue, cortical inh.; cyan, noisy inhib), lower panels Exc. Complex Firing rate vs. input conductance for 4 networks with varying N: 25 (blue), 50 (purple), 100 (green), 200 (red). Hysteresis occurs for N=100 and 200. Constant S/N Left. Fluctuation-controlled transition is sharp in NMDA-AMPA ratio (and in Neff; not shown here). Complex cells in two-population far-ring model at 4 contrasts for networks with different NMDA-AMPA ratios (22.5%, 25% and 27.5%). A mere 2.5% increase in NMDA-AMPA moves the network to a hysteretic (or bistable) state! • Fluctuations in Synaptic Conductances governed by • small N networks, sparse coupling, and synaptic failure • AMPA vs. NMDA (excitation) and GABAA (inhibition) (not shown here) • Analysis of Bifurcations can be done using • Fokker-Planck approach (small N networks) • Kinetic Theory formalism of Cai et al (2004) Bifurcations in the Mean-Driven Limit Conductance-based Integrate-and-Fire Neurons Excitatory and Inhibitory; Specification of coupling strengths Bifurcation in the Two Population (S and C), All-to-All Network with N neurons: S-population receives feed forward excitatory input and recurrent network coupling C-population receives only recurrent network coupling Whenever , the neuron fires and • Applications to Network Models of V1: • 1. A sparsified egalitarian model: • 2. Ring model of Orientation Selectivity: p, connection probability, Sexc,inh synaptic coupling strengths Left. Hysteresis in V1 Model. The distribution of DNspikes (difference in spikes during ramp-up and ramp-down of stimulus contrast) for the exc. Simple (dashed) and exc. Complex (solid) populations in 2 networks: Neff=96 (top), Neff=768 (bottom). The exc. Complex cells in the Neff=768 tend to be hysteretic in stimulus contrast. In the Mean-driven limit, the population firing rates MS and MC obey • Nonspecific and Isotropic (egalitarian) cortical coupling, • Total (LGN + cortical) excitation on a cell is constant Miller 96, Royer & Pare 02 • Combined AMPA and NMDA excitation • Intracortical connections sparse and random (each neuron is connected to N=96 neurons) Thanks to J. Andrew Henrie, G. Kovacic, A. Rangan and support by the NSF DMS-0506396 Average Firing Rate, • References: • Tao, Shelley, McLaughlin & Shapley, PNAS 101, 366 (2004) • Cai, Tao, Shelley & McLaughlin, PNAS 101, 7757 (2004) • Fitzpatrick, Lund & Blasdel, J. Neurosci 5, 3329 (1985) • Callway & Wiser, Vis. Neurosci. 1, 907 (1996) • Miller, Neuron 17, 371 (1996) • Royer & Pare, Neurosci. 115, 455 (2002) • Phase-plane: dM/dtvs.M • dM/dt not monotonic in M • At fixed S and Ginput, there can exist 1, 2 or 3 steady state solutions • Same bifurcation structure in Ginput (with fixed S) S = 0.42 S = 0.385 S = 0.35 Conclusions. The emerging picture of the cortical network shows a state dominated by fluctuations. We have examined a fluctuation-controlled bifurcation, whose distinguishing features include near hysteresis and steep network gain. This bifurcation underlies the contrast invariant orientation selectivity in a ring model. Furthermore, work based on a large-scale network model of V1 suggests that a network operating near this fluctuation-controlled critical state can reproduce many experimentally measured aspects of cortical response, including contrast invariant orientation selectivity. Our theory identifies control parameters of network models to achieve criticality and points to ways to avoid unphysiological bifurcations. S = 0.305 S = 0.25 • A one-dimensional model of V1 • Effective lengthscales of cortico-cortical coupling are functions of ring size (or distance to pinwheel center) Saddle-Node Bifurcations