Imprecise correlated activity in self-organizing maps of spiking neurons

Neural Networks 21 (2008) 810–816

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2008 Special Issue

Imprecise correlated activity in self-organizing maps of spiking neurons Francisco J. Veredas ∗ , Héctor Mesa, Luis A. Martínez Dpto. Lenguajes y Ciencias de la Computación, Universidad de Málaga, Bulevar de Louis Pasteur s/n, 29071, Málaga, Spain

article

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Article history: Received 15 January 2008 Received in revised form 19 May 2008 Accepted 17 June 2008 Keywords: Self-organizing Spiking neurons STDP Correlated activity

a b s t r a c t How neurons communicate with each other to form effective circuits providing support to functional features of the nervous system is currently under debate. While many experts argue the existence of sparse neural codes based either on oscillations, neural assemblies or synchronous fire chains, other studies defend the necessity of a precise inter-neural communication to arrange efficient neural codes. As it has been demonstrated in neurophysiological studies, in the visual pathway between the retina and the visual cortex of mammals, the correlated activity among neurons becomes less precise as a direct consequence of an increase in the variability of synaptic transmission latencies. Although it is difficult to measure the influence of this reduction of correlated firing precision on the self-organization of cortical maps, it does not preclude the emergence of receptive fields and orientation selectivity maps. This is in close agreement with authors who consider that codes for neural communication are sparse. In this article, integrate-and-fire neural networks are simulated to analyze how changes in the precision of correlated firing among neurons affect self-organization. We observe how by keeping these changes within biologically realistic ranges, orientation selectivity maps can emerge and the features of neuronal receptive fields are significantly affected. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The pioneering work of Hubel and Wiesel (1962) on cortical activity in the primary visual cortex of cats showed that cortical neurons were more effectively activated when a certain receptive field on the retina was stimulated. They also demonstrated that the position of the receptive fields, the orientation selectivity and the ocular dominance present a global columnar organization in the cortex. However, there is still some debate on how cortical orientation selectivity is set up (Ferster & Miller, 2000). While diverse physiological (Ferster, Chung, & Wheat, 1996; Reid & Alonso, 1995) and theoretical (Linsker, 1986; von der Malsburg, 1973) studies support the role of feed-forward connectivity for orientation selectivity (see Miller, Erwin, and Kayser (1999) and Swindale (1996) for revisions), other reputable works (Ben-Yishai, Bar-Or, & Sompolinsky, 1995; Martin, 2002; Somers, 1995) point to a greater influence of intracortical circuitry and inhibitory connections. Another controversial issue is how the neurons communicate with each other to form the proper circuitry that will robustly support different features of the central nervous system. While many studies address the possibility of having a sparse neural code based on neural assemblies (Gerstein & Kirkland, 2001),

∗

Corresponding author. Tel.: +34 95213 7155; fax: +34 95213 1397. E-mail addresses: [email protected] (F.J. Veredas), [email protected] (H. Mesa), [email protected] (L.A. Martínez). 0893-6080/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2008.06.006

synchronous fire chains (Gewaltig, Diesmann, & Aertsen, 2001), or oscillations (Ritz & Sejnowski, 1997), others claim the existence of neural codes that require a precise inter-neural communication to be arranged (Reich, Victor, Knight, Ozaki, & Kaplan, 1997; Reinagel & Reid, 2000). Neuronal response properties undergo several important transformations from the retina to the primary visual cortex. Receptive fields become more elaborated (Hubel & Wiesel, 1962), average firing rates are reduced and visual responses become more variable (Kara, Reinagel, & Reid, 2000). Studies of cross-correlation analysis suggest an additional transformation in which the correlated firing between presynaptic and postsynaptic neurons becomes less precise. While retinogeniculate connections generate very narrow correlograms with less than 1 ms width at halfamplitude (Usrey, Reppas, & Reid, 1999), the connections from the lateral geniculate nucleus (LGN) to cortical layer 4 and from layer 4 to layers 2 and 3 generate much wider correlograms (Alonso, Usrey, & Reid, 2001). One of the main factors explaining this reduction of precision of the inter-neural correlated activity is synaptic jitter (Veredas, Vico, & Alonso, 2005). In the pathway between the retina and the visual cortex of mammals, neural synaptic jitter (i.e. the variability of synaptic transmission latencies) increases. For example, in the developing rat neocortex, Markram, Lubke, Frotscher, Roth, and Sakmann (1997) measured fluctuations in excitatory post-synaptic potential latency of 1.5 ms between layer 5 neurons, which is a very large value in comparison with the submillisecond produced by retinogeniculate connections (Cleland, Dubin, & Levick, 1971).

F.J. Veredas et al. / Neural Networks 21 (2008) 810–816

Increasing transmission jitter significantly reduces the precision of the correlated firing (Veredas et al., 2005). The question that arises now is how this reduction of the time accuracy of spike transmissions could affect the organization of feature maps. In this article we analyze the self-organization of receptive fields and orientation selectivity maps in feed-forward networks of integrate-and-fire (IAF) neurons with modifiable connections. Our results show how changes in the precision of the correlated firing among neurons do not prevent the emergence of orientation selectivity maps but affect the features of receptive fields. This means that reducing the exactness of inter-neural communication significantly influences the configuration of receptive fields in this sort of feed-forward network. We modified this accuracy of the correlated activity by introducing incremental and physiologically plausible changes in the synaptic transmission jitter, i.e. increasing the variability of synaptic transmission delays. We address the problem using a spike-time dependent plasticity (STDP) (Panchev & Wermter, 2004; Rao & Sejnowski, 2001) approach, where modifiable excitatory synapses change during a learning process to converge into a self-organizing structure that presents orientation sensitive receptive fields. 2. The spiking Neuron model IAF models are a particular case of simplified neuronal models. The traditional form of an IAF neuron (Stein, 1967) consists of a first order differential equation (Eq. (1)) with a subthreshold integration domain (where the neuron integrates its inputs I (t )) and a threshold potential (not explicitly represented in the equations) for action potential generation. Cm

dVm (t ) dt

= I (t ) −

[Vm (t ) − Vrest ] Rm

,

(1)

where Cm is the neuronal membrane capacitance, Vm the membrane potential, Rm the membrane resistance, Vrest the resting potential and I (t ) the synaptic input current. For the purposes of this paper, I (t ) has been modeled as in Eq. (2). I (t ) =

X

ωj g (tˆj )[Esyn − Vm (t )] + noise,

(2)

j

where ωj represents the connection weight (or synaptic efficacy) between neuron j (presynaptic) and the current neuron (postsynaptic), g (s) is the synaptic conductance, tˆj is the time relative to the last spike of the presynaptic neuron j (also taking into account the existence of a synaptic delay (δ ) and a jitter of transmission delay (ι), modeled as the mean and the standard deviation of a Gaussian distribution of synaptic latency, respectively), Esyn is the reversal potential or synaptic equilibrium and, finally, noise represents the background synaptic noise (following a Gaussian distribution with parameters (µn , σn )). Synaptic conductance has been modeled (Eq. (3)) by the difference of two negative exponential functions with different time-constants, τr and τd , which affect the rise and decay times of the conductance curve, respectively. g (t ) = gˆ e−1/τr − e−1/τd ,





(3)

where gˆ modulates the maximum synaptic conductance. For the computational implementation of the IAF neuron model, Eq. (1) has been integrated by the Backward Euler Integration Method. A time resolution of ∆t = 0.1 ms was chosen to ensure the stability of the system and to allow the introduction of all the physiological phenomena that could contribute to precise correlated firing (Veredas et al., 2005) and could occur within a time interval less than 1 ms (Bloomfield & Sherman, 1988; Eysel, 1976), as is the case of synaptic transmission jitter.

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To complete the neuron model, an absolute refractory period (of 2.5 ms) was included and also an after-hyperpolarization potential value where the membrane potential decays to when an action potential occurs. This after-hyperpolarization potential constitutes an additional relative refractory period (Hodgkin & Huxley, 1952; Nagumo, Arimato, & Yoshizawa, 1962), and it has been assigned an empirical value in our simulations that is calculated as 20% of the membrane potential once it exceeds the threshold potential: Vm (t f ) = 1.2 × Vm (t f − ∆t ), where t f is the spike time. In computer simulations of the neuron model above, synaptic conductances have to be updated at each simulation step, which is a crucial task from the point of view of computational efficiency. The basic problem consists of computing the overall conductance in a more efficient way than simply using the convolution of spike trains with conductance functions of all synapses at each simulation step. Starting from conductance equation (3), the Z-transformation of the discretization of g (t ) was used to obtain a recursive expression of g [n] as a function of only a few previous terms. This recursive function allows us to state each overall conductance value at step n as a function of a reduced number of the previous conductance values and the presynaptic spike trains (Köhn & Wörgötter, 1998), significantly increasing the computational efficiency of the model (Veredas & Mesa, 2006). 3. Precision of correlated activity and synaptic jitter As revealed by physiological and theoretical studies (see Veredas et al. (2005)), increasing the synaptic transmission jitter – defined as the variability of the synaptic delay – affects the precision of the correlated firing in a monosynaptic connection. When this correlated activity is measured by cross-correlation analysis of the activity of two monosynaptically connected neurons, the increase of transmission jitter is evidenced by significant changes in the monosynaptic peak of the correlogram. For the results in Fig. 1 two monosynaptically connected IAF neurons following the model above were simulated. The left graph of the figure shows the correlogram for a simulation with jitter of ι = 0.1 ms, while the jitter for the right graph of the figure was 1.5 ms. As can be observed, changes in the jitter affect the amplitude, position and width of the monosynaptic peak and, consequently, reveal changes in the precision of the correlated firing. In this paper we present the results of simulating feed-forward networks of excitatory IAF neurons with spike-timing plastic synapses and analyze the effects of changing synaptic jitter on the self-organization of receptive fields and also their orientation selectivity. 4. The STDP learning rule Modifiable synapses are modeled by an STDP rule (see Eqs. (4) and (5)). STDP rules model synaptic plasticity by considering in their formulation the precise timing of neuronal activity, so that synaptic efficacies are enhanced or reduced in a way that depends on the time distance of each post- and presynaptic spike. For the network architecture presented in this paper, only excitatory synapses are considered, which are governed by Eqs. (4) and (5) (based on the learning rule for excitatory synapses published by Bartsch and van Hemmen (2001)). W (t ) =



0 e−t /τw

if t < 0 if t ≥ 0,

(4)

where τw is the learning time constant for synaptic modifications that depend on the distance between the post- and the presynaptic

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Fig. 1. Simulation of a monosynaptic connection with two different synaptic jitters. Left: jitter ι = 0.1 ms; position of maximum peak = 2.1 ms; maximum peak amplitude = 143 spikes; width at 50% of maximum peak amplitude = 3.3 ms; width at 25% of maximum peak amplitude = 7.4 ms. Right: jitter ι = 1.5 ms; position of maximum peak = 3.1 ms; maximum peak amplitude = 100 spikes; width at 50% of maximum peak amplitude = 6.6 ms; width at 25% of maximum peak amplitude = 10.6 ms.

spikes. In the interests of simplicity, synaptic depression by negative distances of post- and presynaptic spikes (left half of the learning window) has not been included in the model (Bartsch & van Hemmen, 2001): weight reductions are only modulated by the occurrence of postsynaptic spikes (see Eq. (5)).

   X   f W t − tj + η + ζ − ξ ωj (t ), ∆ωj (t ) = s(t )  

f tj

Table 1 Simulation parameters (A) Neuronal parameters

∆t

τm

Cm

Vrest

Vthres

ϑ

ρ

0.1 ms

1 ms

0.1 nF

−70 mV

−40 mV

0.2

2.5 ms

(B) Synaptic parameters

(5)


where s(t ) is the activity of the postsynaptic neuron at the current time, with s(t ) = 1 if the postsynaptic neuron fires, and s(t ) = 0 if the postsynaptic neuron remains at rest; W is the STDP learning f window; tj is the time step f when the presynaptic neuron emits an action potential; η is a learning rate constant that modulates the amplitude of incoming synaptic weight reductions due to the emission of each postsynaptic spike; ζ represents a sort of activityindependent synaptic enhancement: without activation, a synapse will slowly approach some non-zero efficacy; finally, ξ modulates the reduction of each synapse at a rate proportional to its current weight.

δ

τr

τd

µn

σn

1.0 ms

4.0 ms

1.0 ms

2.7 nA

0.5 nA

(C) Learning parameters

τwi–ii 11 ms

τwii-iii 5 ms

η

ζ

ξ

−0.57

9.5 × 10

−4

1.25 × 10−6

(A) Default physiological parameters for each IAF neuron. τm represents the membrane time-constant; Vthres is the threshold potential; ϑ is an afterhyperpolarization factor; ρ is an absolute refractory period. (B) Synaptic and transmission parameters. (C) Parameters for synaptic learning. (i, ii and iii represent input, intermediate and output layers, respectively.)

5. Simulations and results The network architecture for simulations consists of three twodimensional layers of 32 × 32 IAF neurons with feed-forward modifiable excitatory connections from input to output. There are no lateral connections. Receptive fields of neurons in intermediate and output layers consist of an 11 × 11 array of afferent synapses from the presynaptic layer, centered at the position of the postsynaptic neuron, assuming that layers are circular matrices for neighboring considerations. Synaptic weights are initialized as a function of the distance between each post- (i) and presynaptic √ (j) neuron: wij (0) = λe− (xi −xj ) +(yi −yj ) /σ , where x and y are the row and column of the neuron position in the layer, and λ is a scale factor. Although in real cortex the synaptic efficacy of two connected neurons as a function of their distance could be a random number (Braitenberg & Schüz, 1991), in our simulations however we have introduced this weight initialization depending on the distance which expresses the expected efficacy from neuron j to neuron i (Bartsch & van Hemmen, 2001). In Table 1 default neuronal, synaptic and learning parameters are shown. Neurons in the input layer are stimulated by convolving two-dimensional Gaussian functions with a stationary noise signal that follows a Gaussian distribution (with parameters N (µn , σn )) (see Fig. 2). This random stationary signal generates a basal activity in the network that is large enough to ensure regular 2

2

Fig. 2. Network input stimulation pattern: stationary signals following a Gaussian distribution N (2.7 µA, 0.5 µA) are convolved with a two-dimensional Gaussian function with an amplitude of 0.0847 µA and σ = 4. X and Y axes represent the position of neurons in the input layer.

spiking patterns from the neurons as well as synaptic correlated activity between the layers (Veredas et al., 2005). The convolution with two-dimensional Gaussian functions introduces an activity correlation into the network that allows the emergence of receptive fields and orientation selectivity maps in the absence of lateral connections (Kammen & Yuille, 1988; Linsker, 1986; Miller, 1994; Stetter, Lang, & Müller, 1993; von der Malsburg, 1973; Winbauer, Wenisch, Miller, & van Hemmen, 1997; Winbauer, Wenisch, van Hemmen, & Miller, 1997), as will be shown later.

F.J. Veredas et al. / Neural Networks 21 (2008) 810–816

Neurons in the intermediate and output layers receive, as a stimulation signal, a source of synaptic noise current which is integrated with the presynaptic input from neurons in the preceding layer. Synaptic transmission delay has been modeled (see Section 2) as a Gaussian distribution of synaptic latencies with a mean of δ = 0.1 ms – the same for all simulations – and a standard deviation that represents synaptic jitter (ι). In Fig. 3 the results from three simulations of the feed-forward architecture above are shown. Each row in this figure represents an independent simulation for a different configuration of jitter. In the panels on the left of the figure the orientation selectivity maps for the output layers are represented, which emerge from the selforganization of all synapses in each network. These are modifiable via the STDP rule of Eqs. (4) and (5). Each color square represents one neuron’s orientation selectivity as is indicated in the colour scale bar of the figure (expressed in degrees). As can be observed in these panels, typical arrangements of orientation selectivity bands and singularities emerge. On the other hand, in the panels on the right of Fig. 3, the receptive fields of the neurons in the output layers are visible: a gray-level scale has been used to represent the synaptic weights (0: black; 1: white) in each 11 × 11 receptive field. As can also be observed from these panels, clusters of neurons with similar receptive fields emerge with the self-organizing process. For the simulation of Fig. 3A and B all synapses in the network have a uniform jitter of ι = 0.1 ms, while in Fig. 3C and D, a larger synaptic jitter of ι = 1.5 ms for the whole synaptic tree of the network is used. Finally, in Fig. 3E and F, the results from a simulation of the network architecture with non-uniform jitter are presented. For this latter simulation, jitter was modeled as a two-dimensional inverse Gaussian function with a minimum jitter value at the central synapse at each receptive field of the network. Also an increasing jitter was modeled when going from the center to the periphery of the receptive fields, as it has been formulated in the following equation: −[([x−χ/2]2 +[y−Υ /2]2 )/σ ]

J (x, y) = Γ − %e

,

(6)

where Γ is the maximum likely jitter, % is a scale factor; x and y are the row and column of position of the presynaptic neuron, respectively; χ × Υ = 11 × 11 neurons is the size of the receptive field (matrix dimensions of incoming synaptic weights), and σ is the standard deviation of a Gaussian function. For the simulation in Fig. 3 E and F, Γ = 1.5 ms, % = 1.4 ms and σ = 9, which gave a maximum likely jitter of ι ≈ 1.2 ms at the periphery of each receptive field and a minimum jitter of ι = 0.1 ms at its center. As can be observed in Fig. 3C–F, for larger jitter values orientation selectivity maps and clusters of receptive fields still emerge, as for the smaller jitter values of Fig. 3A and B, by synaptic self-organization, so that a significant reduction in the precision of the correlated firing caused by increasing the jitter (see Section 3) does not dramatically preclude the formation of receptive fields and orientation selectivity maps. In Tables 2 and 3 we have gathered several measurements taken from the results of simulations of the network architecture above with different jitter arrangements. Receptive field clusters (i.e. grouping of receptive fields visually observable) have been identified in each panel (right panels of Fig. 3) by classical image processing techniques (Gonzalez & Woods, 2002). In Fig. 4 an example of cluster segmentation is shown: from the image of synaptic weights (Fig. 4 left), dilation, erosion and opening operators are applied to separate each independent cluster of receptive fields from the others (see Fig. 4 right). In Fig. 4 left, a threshold level was set to consider a receptive field as that formed by those afferent synapses with an efficacy weight ≥ 0.5. Elongation measurements of clusters and receptive fields have been calculated as 1 − 4π · area/perimeter 2 .

813

Table 2 Cluster measurements from simulations of networks with different jitter configurations Uniform jitter 0.1 ms

Gaussian jitter 1.0 ms

1.5 ms

σ =9

σ =1

9.9 1.8 9

12.7 2.1 13

13.2 2.2 14

11.1 1.5 11

12.5 2.2 12

98.1 67.3 87.3

68.6 49.6 61.2

52.9 40.9 42.5

74.5 51.1 65.8

67.7 50.9 61.1

0.5 ms

Number of clusters Mean Standard deviation Median

9.4 2.5 9

Cluster area: number of neurons Mean Standard deviation Median

111.3 81.1 99.9

Cluster elongation (0: circle; 1: line) Mean Standard deviation Median

0.31 0.11 0.30

0.29 0.10 0.27

0.26 0.08 0.25

0.24 0.08 0.23

0.28 0.09 0.27

0.26 0.09 0.25

Each measurement was obtained as an average of 25 simulations with identical parameters (and 50,000 ms of simulation time each). Four former columns show measurements from simulations with uniform jitter for the whole synapses in the net; two later columns show the results for synaptic jitter configured by an inverse Gaussian function (Γ = 1.0 ms; % = 0.9 ms; see Eq. (6)).

Table 3 Receptive field measurements from simulations of networks with different jitter configurations Uniform jitter 0.1 ms

0.5 ms

Gaussian jitter 1.0 ms

1.5 ms

σ =9

σ =1

14.6 4.2 14.7

12.1 4.2 12.4

14.2 4.4 14.3

13.9 4.2 14.1

Receptive field size: number of synapses Mean Standard deviation Median

18.4 4.4 18.4

17.3 4.2 17.5

Receptive field elongation (0: circle; 1: line) Mean Standard deviation Median

0.71 0.16 0.73

0.70 0.16 0.73

0.69 0.17 0.72

0.66 0.17 0.63

0.69 0.17 0.70

0.69 0.17 0.71

Each measurement was obtained as an average of 25 simulations with identical parameters (and 50,000 ms of simulation time each). See Table 2 caption for more details.

As revealed from the results shown in Fig. 3, although increasing jitter (in a uniform or non-uniform manner) does not prevent the self-organization of receptive fields and orientation selectivity maps, significant differences can be found by closely inspecting the size and shape of the receptive fields and their clusters. In Table 2, data from cluster measurements show an increasing number of clusters as well as a decreasing cluster area and elongation as uniform jitter increases (with linear regression coefficients of γ = 0.6, γ = −0.72 and γ = −0.71, respectively). On the other hand, receptive field measurements (see Table 3) show significant differences in both the size and elongation of receptive fields: as uniform jitter increases, the size and the elongation of the receptive fields decrease linearly (with linear regression coefficients of γ = −0.93 and γ = −0.72, respectively). These tendencies of cluster and receptive field degeneration would seriously impede synaptic self-organization and the formation of orientation selectivity maps for larger non-realistic jitter values. (Our results for uniform jitter larger than 1.5 ms addressed this issue, although these results are not shown here.) For Gaussian jitter (non-uniform distribution of jitter into the receptive field), changes in the standard deviation (σ ) analyzed in our simulations confirm these results, as can be seen in the two last columns of Tables 2 and 3. Finally, we have not observed significant differences in statistics of the distributions of orientation selectivity in maps with different jitter arrangements for the physiological values of jitter managed in our simulations. The parameters of Table 1 were empirically set

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Fig. 3. Orientation selectivity maps (A, C, E) and receptive fields (B, D, F) of feed-forward synapses from the intermediate layer to the output layer for three different configurations of jitter (simulation time = 50,000 ms). Each row represents a different simulation. See main text for details.

to ensure convergence to significant network states. Changes in those parameters may drive the network to trivial or oscillating states. 6. Conclusions We analyzed the effects of increasing the jitter of transmission delay on the self-organization of receptive fields and orientation selectivity maps. For this purpose, we simulated feed-forward

networks of IAF excitatory neurons with modifiable synapses – by an STDP learning rule – and determined how a reduction in the precision of the correlated activity affects the final arrangement of neural receptive fields. Our main results show how when the precision of the interneural correlated activity is reduced, the mean size and elongation of receptive fields in the output layer decrease linearly. Although the increasing variability of neural responses across the visual pathway could affect the shape of receptive fields, as

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Fig. 4. An example of cluster segmentation. The left image shows the final afferent synaptic weights from a simulation with homogeneous jitter (ι = 0.1 ms). The right image shows the results from the application of dilation, erosion and opening operators on the left image to separate and identify each cluster of receptive fields: for representative purposes each independent cluster has been assigned a different pseudo-color.

our results indicate, the measured physiological magnitude of this variability seems not to be a determinant factor in preventing the emergence of self-organizing orientation selectivity maps. If synaptic jitter values remain within realistic physiological ranges, these changes in the shape of receptive fields do not preclude the self-organization of orientation selectivity maps in this sort of excitatory feed-forward architecture. These results are in close agreement with works claiming the unnecessary sharpness of time precision of interneural spike transmission for the formation of some sort of emerging neural features (Gerstein & Kirkland, 2001; Gewaltig et al., 2001; Ritz & Sejnowski, 1997) such as orientation selectivity. Acknowledgments This research has been technically supported by the Grupo de Estudios en Biomimética of the Universidad de Málaga, Spain, and partially funded by the Consejería de Salud, project PI0197/2007, and the Consejería de Innovación, Ciencia y Empresa, PAI, project P06-TIC01615, Junta de Andalucía, Spain. References Alonso, J.-M., Usrey, W., & Reid, R. (2001). Rules of connectivity between geniculate cells and simple cells in cat primary visual cortex. Journal of Neuroscience, 21(11), 4002–4015. Bartsch, A., & van Hemmen, J. (2001). Combined Hebbian development of geniculocortical and lateral connectivity in a model of primary visual cortex. Biological Cybernetics, 84(1), 41–55. Ben-Yishai, R., Bar-Or, R., & Sompolinsky, H. (1995). Theory of orientation tuning in visual cortex. Proceedings of National Academy of Sciences, USA, 92, 3844–3848. Bloomfield, S., & Sherman, S. (1988). Postsynaptic potentials recorded in neurons of the cat’s lateral geniculate nucleus following electrical stimulation of the optic chiasm. Journal of Neurophysiology, 60, 1924–1945. Braitenberg, V., & Schüz, A. (1991). Anatomy of the cortex. Berlin, Heidelberg, New York: Springer. Cleland, B., Dubin, M., & Levick, W. (1971). Sustained and transient neurones in the cat’s retina and lateral geniculate nucleus. Journal of Physiology, 217, 473–496. Eysel, U. (1976). Quantitative studies of intracellularpostsynaptic potentials in the lateral geniculate nucleus of the cat with respect to optic tract stimulus response latencies. Experimental Brain Research, 25, 469–486. Ferster, D., Chung, S., & Wheat, H. (1996). Orientation selectivity of thalamic input to simple cells of cat visual cortex. Nature, 380, 249–252. Ferster, D., & Miller, K. (2000). Neural Mechanisms of Orientation Selectivity in the Visual Cortex. Annual Reviews of Neuroscience, 23, 441–471. Gerstein, G., & Kirkland, K. (2001). Neural assemblies: Technical issues, analysis, and modeling. Neural Networks, 14, 589–598. Gewaltig, M.-O., Diesmann, M., & Aertsen, A. (2001). Propagation of cortical synfire activity: Survival probability in single trials and stability in the mean. Neural Networks, 14, 657–673.

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