Levels in Computational Neuroscience Reasonably good understanding (for our purposes!) Poor understanding Poorer understanding Very poorer understanding

From neuron to network

The layered structure of the first visual area, and connections to other areas (Fig in Kandel and Schwartz, Principles of Neural Science)

The columnar organization of the monkey visual cortex (Fig in Shepherd, The Synaptic Organization of the Brain)

Definition of the firing rate in terms of a temporal average. (Fig. 1.9, Spiking Neuron Models)

Definition of the firing rate in terms of the peri-stimulus-time- histogram (PSTH) as an average over several runs of an experiment. (Fig. 1.10, Spiking Neuron Models)

Definition of the firing rate as a population density. Gerstner & Kistler Fig. 1.11

Feedforward inputs to a single neuron. Dayan and Abbott Fig. 7.81

Feedforward and recurrent networks Dayan and Abbott Fig. 7.3

Dayan and Abbott Fig. 7.4 Coordinate transformations during a reaching task Target Fixation Gaze angle Retinal angle Body coordinates Objective: transform from retinal coordinates to body coordinates

Tuning curves of a visually responsive neuron in premotor cortex Dayan and Abbott Fig. 7.5 Head fixed Fixate on Body coordinates Response curve fixed! Retinal coordinates Curve shifts to compensate! Head rotates Fixation fixed Model tuning curve g=0 0 g=10 0 g=-20 0

Dayan and Abbott Fig. 7.6 The gaze-dependent gain modulation of visual responses of neurons in area 7a Tuning curve 2 Gaze directions Gaze independence! Related to s 2D tuning function

burst and an integrator neurons involved in horizontal eye positioning Dayan and Abbott Fig. 7.7

Eigenvector expansion

Steady state rates – linear network Real-valued matrix M: use real and imaginary parts

Selective amplification by a linear network Dayan and Abbott Fig. 7.8 Input: cosine with peak at  = 0 o + added noise Fourier amplitude of inputs Output: steady state Fourier amplitude of output  = 0 component enhanced All Fourier components present

Effect of nonlinearity on amplification Dayan and Abbott Fig. 7.8 Smoother response Several Fourier components appear

Visual information flow Dayan and Abbott Fig.2.5 Center surround response Oriented response

Visual receptive fields Dayan and Abbott Fig Mathematical fit Actual response LGN neuron Center surround Orientation selective V1 neuron (simple)

Hubel Wiesel model Low response Simple summation Vertical responseUndirected response High response

Effect of contrast Dayan and Abbott Fig input contrast levels Note: response is amplified but Real responses Network amplification not broadened

Nonlinear winner-takes-all selection Dayan and Abbott Fig Input: cantered at ±90 0 Output: Higher peak selected

Associative recall Dayan and Abbott Fig representative unitsMemory: units high, others low Memory: every 4 th unit high N v =50, 4 patterns Partial inputsConverged outputs

Pattern recall – Hopfield model InputOutput Time

Dayan and Abbott Fig Excitatory-Inhibitory network NullclinesEigenvalues Unstable Stable

Dayan and Abbott Fig Excitatory-Inhibitory network Temporal behavior Stable fixed point

Dayan and Abbott Fig Excitatory-Inhibitory network Temporal behavior Unstable fixed point – limit cycle

Dayan and Abbott Fig Extracellular field potential in olfactory bulb Olfactory model I To cortex Excitatory Inhibitory interneurons Sniffs Oscillatory neural activity No fast oscillations

Dayan and Abbott Fig Olfactory model II Activation functionsEigenvalues Region of instability

Dayan and Abbott Fig Olfactory model III Behavior during a sniff cycle Identity of odor determined by: Amplitudes and phases of oscillations Identity of participating mitral cells