Figure 2:

*The Golden Laplacian Pyramid.*To represent the edges of the image at different levels, we may use a simple recursive approach constructing progressively a set of images of decreasing sizes, from a base to the summit of a`pyramid`*. Using simple down-scaling and up-scaling operators we may approximate well a Laplacian operator. This is represented here by stacking images on a*, that is where the aspect ratio is the golden section $\phi \eqdef \frac{1+\sqrt{5}}{2}$. We present here the base image on the left and the successive levels of the pyramid in a clockwise fashion (for clarity, we stopped at level $8$). Note that here we also use $\phi^2$ (that is $\phi+1$) as the down-scaling factor so that the resolution of the pyramid images correspond across scales. Note at last that coefficient are very kurtotic: most are near zero, the distribution of coefficients has`Golden Rectangle``long tails`*.*Figure 1:

**Band-pass motion stimuli for perception and action tasks.***(a)*In the space representing temporal against spatial frequency, each line going through the origin corresponds to stimuli moving at the same speed. A simple drifting grating is a single point in this space. Our moving texture stimuli had their energy distributed within an ellipse elongated along a given speed line, keeping constant the mean spatial and temporal frequencies. The spatio-temporal bandwidth was manipulated by co-varying Bsf and Btf as illustrated by the (x,y,t) examples. Human performance was measured for two different tasks, run in parallel blocks.*(b)*For ocular tracking, motion stimuli were presented for a short duration (200ms) in the wake of a centering saccade to control both attention and fixation states.*(c)*For speed discrimination, test and reference stimuli were presented successively for the same duration and subjects were instructed to indicate whether the test stimulus was perceived as slower or faster than reference.Figure 3:

**Equivalent MC representations of some classical stimuli.***(A, top)*: a narrow-orientation-bandwidth Motion Cloud produced only with vertically oriented kernels and a horizontal mean motion to the right.*(Bottom)*: The spectral envelopes concentrated on a pair of patches centered on a constant speed surface. Note that this`speed plane" is thin (as seen by the projection onto the ($f_x$,$f_t$) face), yet it has a finite thickness, resulting in small, local, jittering motion components.`*(B)*a Motion Cloud illustrating the aperture problem.*(Top)*: The stimulus, having oblique preferred orientation ($\theta=\frac{\pi}{4}$ and narrow bandwidth $B_{\theta}=\pi/36$) is moving horizontally and rightwards. However, the perceived speed direction in such a case is biased towards the oblique downwards, i.e., orthogonal to the orientation, consistently with the fact that the best speed plane is ambiguous to detect.*(C)*: a low-coherence random-dot kinematogram-like Motion Cloud: its orientation and speed bandwidths, $B_{\theta}$ and $B_{V}$ respectively, are large, yielding a low-coherence stimulus in which no edges can be identified.Figure 3:

**Conditions for signal propagation through a feedforward network with correlated inhibition.***(A)*: Model of a feedforward network with correlated inhibition induced by FFI.*(B)*, Signal propagation of a synchronous input through the network when each group projects only onto the RS population of the following group. Due to the instability of the ground-state of purely excitatory feedforward networks (Tetzlaff et al., 2002), transient random fluctuations in the asynchronous background ac- tivity may occasionally induce spontaneously propagating synchrony, as can be observed here some 50 ms before the stimulus onset.*(C)*, Propagation of an asynchronous input through the same network. The asynchronous input induced elevated firing rates in the first groups. However, the activity rapidly synchro- nized over subsequent groups.*(D)*, Propagation of synchronous input was hardly affected by correlated inhibition, induced by including the FS neurons in the target population of the successive group. E, FFI in the feedforward network prevented asynchronous inputs from inducing synchronous activity in subsequent groups.

Figure 3: Representative network simulations (I)–(IX): raster plot sections (top) with corresponding firing rates (middle, last 1.5 s of simulation time), and ISI probability density distributions (bottom, semilogarithmic plots with a zoom-in with linear axes). In the raster plots, gray dots represent inh. spikes, and black dots represent exc. spike times while the other two plots do not distinguish between exc. and inh. neurons. The plots are ordered and numbered according to their occurrence in the phase space: On the top left synchronous regular firing dominates due to low inhibition (small g) and high input rates. On the bottom right, there is asynchronous irregular firing due to a higher inhibition (large g) and lower input rates.

Figure: Most studies of cortical network dynamics are either based on purely random wiring or neighborhood couplings, e.g., [Kumar, Schrader, Aer tsen, Rotter, 2008, Neural Computation 20, 1--43]. Neuronal connections in the cortex, however, show a complex spatial pattern composed of local and long-range connections, the latter featuring a so-called patchy projection pattern, i.e., spatially clustered synapses [Binzegger, Douglas, Martin, 2007, J. Neurosci. 27(45), 12242--12254]. The idea of our project is to provide and to analyze probabilistic network models that more adequately represent horizontal connectivity in the cortex. In particular, we investigate the effect of specific projection patterns on the dynamical state space of cortical networks. Assuming an enlarged spatial scale we employ a distance dependent connectivity that reflects the geometr y of dendrites and axons. We simulate the network dynamics using a neuronal network simulator NEST/PyNN. Our models are composed of conductance based integrate-and-fire neurons, representing fast spiking inhibitor y and regular spiking excitator y cells. In order to compare the dynamical state spaces of previous studies with our network models we consider the following connectivity assumptions: purely random or purely local couplings, a combination of local and distant synapses, and connectivity structures with patchy projections. Similar to previous studies, we also find different dynamical states depending on the input parameters: the external input rate and the numerical relation between excitatory and inhibitory synaptic weights. These states, e.g., synchronous regular (SR) or asynchronous irregular (AI) firing, are characterized by measures like the mean firing rate, the correlation coefficient, the coefficient of variation and so forth. On top of identified biologically realistic background states (AI), stimuli are applied in order to analyze their stability. Comparing the results of our different network models we find that the parameter space necessary to describe all possible dynamical states of a network is much more concentrated if local couplings are involved. The transition between different states is shifted (with respect to both input parameters) and sharpened in dependence of the relative amount of local couplings. Local couplings strongly enhance the mean firing rate, and lead to smaller values of the correlation coefficient. In terms of emergence of synchronous states, however, networks with local versus non-local or patchy versus random remote connections exhibit a higher probability of synchronized spiking. Concerning stability, preliminary results indicate that again networks with local or patchy connections show a higher probability of changing from the AI to the SR state. We conclude that the combination of local and remote projections bears important consequences on the activity of network: The apparent differences we found for distinct connectivity assumptions in the dynamical state spaces suggest that network dynamics strongly depend on the connectivity structure. This effect might be even stronger with respect to the spatio-temporal spread of signal propagation.

Figure 2 To model spatial integration in the OFR in primates (humans and macaques), we use the tools of statistical inference with the hypothesis that information is represented in a

*probabilistic*fashion. The architecture of the OFR system consists in this model of a stage extracting from the raw image the possible local translation velocities (V1) to represent the local probabilities of translational velocity (MT). These local bits of information are then pooled (MST) to give a single probabilistic representation of possible translational velocities to the oculomotor system, which then controls the eyes' motion as quickly and efficiently as possible to stabilize the image on the retina. The local probabilities may be often described as gaussian probabilities, and the gain response as a function to the signal to noise ratio is then given by a Naka-Rushton curve (Barthélemy et al., 2007).Figure 1

**Basic properties of human OFR.**Several properties of motion integration for driving ocular following as summarized from our previous work. (a) A leftward drifting grating elicits a brief acceleration of the eye in the leftward direction. Mean eye velocity proﬁles illustrate that both response amplitude and latency are affected by the contrast of the sine-wave grating, given by numbers at the right-end of the curves. Quantitative estimates of the sensori-motor transformation are given by measuring the response amplitude (i.e. change in eye position) over a ﬁxed time window, at response onset. Relationships between (b) response latency or (c) initial amplitude and contrast are illustrated for the same grating motion condition. These curves deﬁne the contrast response function (CRF) of the sensori-motor transformation and are best ﬁtted by a Naka–Rushton function (reprinted from (Barthélemy et al., 2007)). (d) At ﬁxed contrast, the size of the circular aperture can be varied to probe the spatial summation of OFR. Clearly, response amplitude ﬁrst linearly grows up with stimulus size before reaching an optimal size, the integration zone. For larger stimulus sizes, response amplitudes are lowered (reprinted from (Barthélemy et al., 2006)). (e) OFR are recorded for center-alone and center–surround stimuli. The contrast of the center stimulus is varied to measure the contrast response function and compute the contrast gain of the sensori-motor transformation at both an early and a late phase during response onset. Open symbols are data obtained for a center-alone stimulus, similar to those illustrated in (c). When adding a ﬂickering surround, ones can see that late (but not early) contrast gain is lowered, as illustrated by a rightward shift of the contrast response function (Barthélemy et al., 2006).Figure 12: Schematic structure of the two-pathway Bayesian model implemented in Barthélemy (2007). The three computational steps to extract parallel, distributed representations of 1D and 2D motion cues are illustrated for a plaid (left-hand panels) and a barberpole (right-hand panels). (a) Outputs of the initial filtering stage extracting 1D and 2D motion features. (b) Likelihood distributions in the velocity space (vx, vy) are plotted for each pathway, together with a common prior distribution centered at (0, 0). (c) A posteriori probability distributions of 1D and 2D motion signals at both low (10%, top row) and high (40%, bottom row) contrast. (Cover from

*Vision Research*, Volume 48, Issue 4, February 2008)Figure:

**An example raster plot of the retina model**We show here the simulated response of ON (blue) and OFF (red) ganglion cells to the presentation of a central spot in a simple model of a retina. Time is in ms, retina grid is 20 by 20. Implemented using PyNN and NeuroTools.Figure 1:

**The log-Gabor transform compared to other multiresolution schemes.***a.*Schematic contours of the log-Gabor filters implented in Fischer (2007) in the Fourier domain with 5 scales and 8 orientations (only the contours at 78% of the filter maximum are drawn).*b.*The real part of the corresponding filters is drawn in the spatial domain. The two first scales are drawn at the bottom magnified by a factor of 4 for a better visualization. The different scales are arranged in lines and the orientations in columns. The low-pass filter is drawn in the upper-left part.*c.*The corresponding imaginary parts of the filters are shown in the same arrangement. Note that the low-pass filter does not have imaginary part. Insets (b) and (c) show the final filters built through all the processes described in Section II.*d.*In the proposed scheme the elongation of log-Gabor wavelets increases with the number of orientations nt . Here the real parts (left column) and imaginary parts (right column) are drawn for the 3, 4, 6, 8, 10, 12 and 16 orientation schemes.*e.*As a comparison orthogonal wavelet filters ’Db4’ are shown. Horizontal, vertical and diagonal wavelets are arranged on columns (low-pass on top).*f.*As a second comparison, steerable pyramid filters [30] are shown. The arrangement over scales and orientations is the same as for the log-Gabor scheme.Figure1: Schematic structure of the primary visual cortex implemented in Fischer (2007). Simple cortical cells are modeled through log-Gabor functions. They are organized in pairs in quadrature of phase (dark-gray circles). For each position the set of different orientations compose a pinwheel (large light-gray circles). The retinotopic organization induces that adjacent spatial positions are arranged in adjacent pinwheels. Inhibition interactions occur towards the closest adjacent positions which are in the directions perpendicular to the cell preferred orientation and toward adjacent orientations (light-red connections). Facilitation occurs towards coaligned cells up to a larger distance (dark-blue connections).

Figure 1:

**Effects of size of a disk grating on the monkey's OFR response.**We present here the gain of the oculo-motor response to a central grating (temporal frequency 10 Hz) as a function of its diameter for the macaque monkey (open circles) and the model (continuous line). Fits were performed as a function of the diameter*d*thanks to equations derived in the text.*(Left)*At low frequencies (0.12~cpd) and contrasts, the gain increases monotonically with the diameter. The curves are well fitted by the simpler equation with only an excitatory pooling.*(Right)*However in more general conditions (here 0.7~cpd), the initial gain decreases after a given diameter suggesting a suppressive effect. This corresponds to a surround inhibition which is well captured by the equation, the inhibition being more pronounced when contrast is higher and contrary to intuition, the diameter $\omega$ extracted from the fits remains constant across curves. This provided a general explanation of the ratio-Of-Guassian model in a probabilistic framework.Figure 1:

**Implementation of the greedy pursuit using Integrate-and-Fire Neurons.**We show the raster plot of 16 neurons tuned for different orientations for the feed-forward (black bars) and the sparse spike coding (white bars) models during the first 150 ms. We simulated here the activity of a network of Integrate-and-Fire neurons tuned to form a simple model of an hyper-column in the primary visual area (V1) to the presentation of a horizontal edge at t=0. In thz sparse coding model, the correlation linked to the information already detected is propagated as a hyper-polarizing and shunting lateral interaction to the neighboring neurons: the response in both latency and spiking frequency to the oriented edge is clearly more selective.

Figure 1:

**Inverse-mapping as a a goal for sensory neural coding.**The visible world is modeled as the interaction of a large set of hypothetical physical sources (world model) according to a known model of their interactions ("synthesis"). We will consider that for sensory cortical areas, the goal of the neural representation (and its implementation by the*neural code*) is to analyze the signal so as to recover at best and as quickly as possible the sources that generated the signal ("analysis") . The analysis may thus be considered as an inverse mapping of the synthesis. A proposed solution for this problem is to*infer*at best the most probable hidden state.Figure 1:

**Progressive reconstruction of the spiking image in the primary visual cortex.**To illustrate that the visual information is contained in the spike code, we show the theoretical reconstruction of the Tiger image using the algorithm presented in the paper. The different edges are extracted using a sparse coding scheme that grabs most salient information first. This reconstruction would correspond to the reconstrucion of the image in an afferent area using the spiking information only. This particular reconstruction on the 256x256 image used a Steerable pyramid with 8 different orientations as the linear transform. The theoretical compression rate compares to JPEG at slow bpp Fig. 2 and is more efficient than the retina model (compare with Lena).Figure 2

**Regularity of edge coefficients distribution in natural images.**We plotted for 200 natural images the mean and variance (depicted by the dotted lines representing the mean $\pm$ one standard deviation) of the decrease of coefficients values in the probabilistic Matching Pursuit scheme described above as a function of the rank. Assuming that these values represent the edge content of the images, it shows that the probability distribution of edge coefficients is regular in natural images and may be used in an efficient compression scheme. It permits to evaluate the coefficients quickly as a function of the rank, and since the transmission error being proportional to the variance to transform efficiently an analog image in a wave of spikes. It should be noted that this decrease is much more rapid than the one observed in the model retina the coefficients are below .15 after 1\% of relative rank.Figure 3:

**Is the spike representation over-complete in the retina?***(Left)*We compared the progressive transmission of information for different degrees of over-completeness in the retina by plotting the average MSE of the residual as a function of the information to code the spike list (in logarithmic scale, propagation up to $12.5\%$ of the relative rank for clarity). The set of neurons used rotation symmetric Mexican hat filters, with scales from layer to layer growing as $\rho=\{ 2,\sqrt{2 },\sqrt[4]{2 },\sqrt[8]{2 } \}$ (and denoted on the legend respectively as 1, 2, 4 and 8). As a comparison we plotted the method used in~\citep{van-Rullen01a} (line 'Wav'). As a function of rank, the MSE decreases more rapidly for increasing degrees of over-completeness.*(Right)*But if we plot the trade-off of MSE with CPU usage as a function of the over-completeness, we find that for the same amount of information the adaptive dyadic strategy is optimal. One should note that the results of the method described in the text is better than the wavelet method of [van-Rullen, 01] since it is adaptative.Figure 1 : Introducing lateral connections between filters in a model of learning (see Publications/Perrinet02sparse) in favor of neighboring activity, we observe the emergence of topological relations between neighboring filters (cyclically numbered from 1 to 16 from left to right and then from bottom to down) similar to the neurophysiologically observed

*pinwheels*.Figure 1

**Progressive reconstruction of the spiking image in the retina.**To illustrate that the visual information is contained in the spike code, we show the theoretical reconstruction of the Lena image using the algorithm presented in the paper. This particular reconstruction on the 256x256 image used a Laplacian pyramid as the linear transform because this transform is invertible and exhibits only little cross-correlation between filters. Results are improved compared to the use of the Calderon frormula (as in [VanRullen, 01]), see Fig. 2 and we recognize the original image after only a few hundreds spikes.Figure 2:

*(A) Look-Up-Table : mean and variance of the absolute contrast in function of the relative rank for a database of 100 natural images (B) Comparison of MSE in function of the file size (in ) for the different methods : (a) MP; (b) MP with Look-Up-Table; (c) JPEG at different qualities.*Figure 1:

**Neural Model of adaptative synchrony detection using STDP (Perrinet, 2002)**:*(left)*Input spikes (with a synfire pattern at t =25ms), are*(middle)*modulated in time and amplitude forming postsynaptic current pulses and are finally*(right)*integrated at the soma. When the potential (plain line) reaches the threshold (dotted line), a spike is emitted and the potential is decreased. A sample PSP (at synapse 1) is also shown.Figure 3: Coherence detection in (Perrinet, 2002) : (left) different input patterns (t = 100ms, 300ms, 500ms, 700ms, 900ms) are (right) learnt by the system: only one neuron per input fires (100 learning steps).