Modeling binocular competition through Hebbian plasticity and constrained connectivity

Neurocomputing 121 (2013) 195–206

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Modeling binocular competition through Hebbian plasticity and constrained connectivity Moran Furman n Department of Neurology, Yale University School of Medicine, New Haven, CT 06520, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 12 April 2012 Received in revised form 29 March 2013 Accepted 21 April 2013 Available online 1 May 2013

In the mammalian brain, inputs from the two eyes compete with each other during development for synaptic contacts with postsynaptic targets. Traditionally, binocular competition has been modeled using a Hebbian-like plasticity rule together with a fixed-sum constraint on total connectivity. Previous work has shown the certain, but not all, formulations of the constrained-connectivity model lead to binocular competition. These differences between model formulations had been analyzed using a correlationbased approach, particularly relevant for the developmental period after eye opening, when the two eyes are normally active together but at varying levels of inter-ocular correlation. More recently, experimental work has established a role for Hebbian plasticity in binocular competition also prior to eye opening. In contrast to later developmental stages, at this pre-vision stage, activity of the two eyes is largely decorrelated. The goal of the present study is twofold. First, a simplified analysis of the constrainedcompetition model is proposed for the case of de-correlated binocular activity as observed prior to eye opening. This analysis provides a more intuitive explanation for the sensitivity of the constrainedconnectivity model to formulation details. Second, based on recent experimental findings, this work proposes a new, modified model in which binocular competition increases the total connectivity instead of operating under a fixed-sum constraint. Simulations and analysis show that this form of “growthpromoting” competition is less sensitive to formulation details and may therefore be biologically advantageous at early stages of visual development. & 2013 Elsevier B.V. All rights reserved.

Keywords: Hebbain model Map development Eye-specific segregation Visual system Retinal waves Dynamical systems analysis

1. Introduction In most mammals, the visual field is divided into two parts: a monocular region, captured by one eye only, and a binocular region, captured by both eyes. In humans, binocular vision covers a major fraction of the visual field and supports important functions such as binocular depth perception and eye-hand coordination [1,2]. Therefore, clarifying the mechanisms of binocular map formation during development has been a major goal of both experimental and theoretical studies [3–5]. Inputs from the two eyes to the brain are physically mixed early in development, and then segregate into eye-specific layers, columns or domains. This process of eye-specific segregation involves activity-dependent competition between afferents from the two eyes. Indeed enhancing or inhibiting activity in one eye (e. g. through visual deprivation) promotes strengthening and expansion of inputs from the more active eye at the expense of the opposite eye (e.g., [4,6]). Whereas after eye opening binocular competition is mediated by visual inputs, prior to eye opening, binocular competition is supported by spontaneous activity patterns that propagate across

n

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0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.04.004

each retina (“retinal waves”) [7–10]. In difference from external visual inputs, which are basically continuous in time, spontaneous retinal waves are characterized by short (2–8 s long) bursting periods of activity in retinal ganglion cells (RGCs), separated by extended periods of quiescence (roughly 40–90 s long; [7,8]). Furthermore, since retinal waves arise spontaneously, most retinal waves are initiated autonomously in each eye. Thus, spontaneous retinal activity is largely de-correlated between the two eyes, with one eye only active at a time. De-correlated inter-ocular activity, as observed prior to eye opening, is ideally suited for supporting segregation between inputs from the two eyes in target brain regions, specifically the thalamic dorsal lateral geniculate nucleus (dLGN) and the midbrain superior-colliculus (SC). Experiments where retinal activity was manipulated pharmacologically, genetically, or optogenetically provide direct support for the role of Hebbian-like, activitymediated binocular competition in the formation of eyesegregated domains in the dLGN and SC prior to eye opening [6,11,12]. Hebbian mechanisms are also likely involved in additional aspects of map formation in the dLGN and SC such as multisensory integration [13–16]. A well-established model of binocular competition is the constrained-connectivity model. In this model, connectivity strength of each eye to its target-neurons is potentiated during correlated pre- and post-synaptic activity [17,18]. Additionally, the

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M. Furman / Neurocomputing 121 (2013) 195–206

summed-connectivity of the two eyes is maintained fixed (i.e., normalized) to prevent unlimited growth of connectivity [19–22]. This normalization constraint could reflect, for instance, homeostatic regulation of synaptic strength (e.g., synaptic scaling, [23,24]). The constrained-connectivity model has been successfully used for analyzing the formation of ocular-dominance column (ODC) in the visual cortex after eye opening (e.g., [5,25–28]). In particular, a number of studies examined how the level of inter-ocular correlation affects the spatial properties of ODC [29–31]. Somewhat disconcertingly however, it was also found that the constrainedconnectivity model is quite sensitive to formulation details. Specifically, the model produces binocular competition under certain, but not all, plasticity rules or types of normalization constraint. This suggests that perhaps binocular competition dynamics as represented in the constrained-connectivity is not best suited to the early and critical stages of visual development, prior to eye opening, which require mechanisms that are robust to external and internal perturbations. As mentioned, the constrained-connectivity model shows substantial sensitivity to formulation details [19,21,22]. This sensitivity has been previously analyzed using a correlation based approach which is particularly suitable for post-eye-opening development. The results of these studies are mathematically solid, but the relatively intricate analysis required in the correlation-based approach limits to some extent an intuitive understanding of the underlying processes. Therefore, examining the constrained-connectivity model from additional perspectives would be beneficial for both modelers and experimentalists. The first goal of this paper is to take advantage of the simpler scenario of binocular competition prior to eye-opening and propose a more direct and intuitive explanation for the sensitivity of the constrained-connectivity model to formulation details. Prior to eye opening retinal activity mostly alternates between the two eyes, making the constrained-connectivity model more amenable to mathematical analysis. The second goal of the paper derives

from recent experimental findings regarding binocular competition prior to eye opening. Patch-clamp recordings from the superior colliculus (SC) of neonatal mice have shown that at this early developmental stage, binocular competition appears to enhance overall synaptic connectivity to target neurons rather than operate under a fixed-sum type of constraint [32]. I therefore propose here a modified version of the constrained-connectivity model in which Hebbian-based binocular competition causes a gradual increase in total connectivity. By applying the analysis technique presented in the first part of the paper, it is shown that this form of “growthpromoting” competition is more robust to variations in formulation details, and therefore represents a form of binocular competition which may be more adequate for early stages of visual development.

2. Methods Model formulation and Hebbian plasticity rule: Two variables, wL(t) and wR(t), represent connectivity from the two eyes to a target neuron (Fig. 1A). Because “retinal waves” (spontaneous activity patterns in the developing retina) are characterized by brief periods of neural bursting separated by long periods of quiescence, as a first approximation, each retinal wave can be considered as a discrete event activating one of the two eyes. Accordingly, simulations and analyses were performed in the discrete domain, advancing one time step at a time. At each time step, connection weights were modified according to a Hebbian plasticity rule Δwi ¼ αρðxi Þsðwi Þy where α is the learning rate, xi the activity level of the corresponding eye, and y is the postsynaptic activity, y ¼ ΦðxL wL þ xR wR Þ The functions ρ and s determine the specific type of plasticity rule been used, and Φ determines the input–output transfer function of

Right eye input

xR

wR y

xL

wL

Target neuron

Left eye input

Divisive normalization

Subtractive normalization

wL 0.8

Connectivity weight

Connectivity weight

1 wR

0.6 0.4 0.2

1

wL

0.8

wR

0.6 0.4 0.2 0

0 0

200

400

600

Time step

800

1000

0

200

400

600

800

1000

Time step

Fig. 1. Model schematic and simulation examples. (A) Model Schematic. Inputs from the two eyes converge onto a target neuron. At each time step, one of the two eyes is activated (vertical lines, left). Connectivity weights (wL,wR) are adjusted according to a Hebbian plasticity rule, and normalized to maintain a fixed sum (wL+wR ¼1). xL,xR indicate the presynaptic activity levels, and y is the postsynaptic activity. (B) and (C) Examples of simulations of a linear Hebbian plasticity rule, with divisive (B) or subtractive (C) normalization. As previously shown, for this plasticity rule, subtractive, but not divisive normalization generates competitive dynamics in which the two connectivity weights diverge.

M. Furman / Neurocomputing 121 (2013) 195–206

wL þ wR ¼ 1 as follows (cf. Mathematic Derivations 1 and 2): In subtractive normalization, half of the total weight increment was subtracted from each of the two connection weights; and in divisive normalization, the two weights were divided by their sum. Weight adjustment in the “growth-promoting” competition model: At each time step, after updating the connectivity weights according to the plasticity rules, the two weights were adjusted so that their sum is increased by a factor β compared to their sum in the previous time step. In subtractive weight-adjustment, an equal term was subtracted from the two weights to achieve the necessary increase in summed-connectivity:

L−eye activated 0.8

The value of β used in simulations was 1.0002. Eye-segregation score: An eye-segregation score, quantifying how much wL(t) and wR(t) diverge from the initial point (0.5,0.5), was defined as follows:
w ðtÞ−w ðtÞ


L R SðtÞ ¼

wL ðtÞ þ wR ðtÞ Analysis of segregation-time as a function of γ (Figs. 5B and 8): Segregation time was calculated for γ values between 0 and 1 at 0.01 incremental steps. Each data point was calculated as average segregation time over 1000 simulations. To calculate the point of minimal slope, the curves was first smoothed using a moving average with span 9, and then the derivative was calculated using a first order finite difference method and similarly smoothed. Programming: The model was implemented in Matlab (MathWorks, MA).

0.4

R−eye activated

0.2

0

0

0.2

0.4

0.6

0.8

1

wL 1

L−eye activated

ðβ−1Þ ∑ðwprev ; wprev Þ L R 2

0.8

wL+ΔwL

and in divisive weight-adjustment, the two weights were scaled to obtain the necessary increase in summed-connectivity: ðwnew ; wnew prev L R Þ ; wprev Þ ðwL ; wR Þ ¼ β R new Þ ∑ðwL ; w ∑ðwnew L R

0.6

start

0.6

end

0.4

R−eye activated

0.2

0

0

0.2

0.4

0.6

0.8

1

wL 0.01

ΔwL following pair of L−R activations

ðwL ; wR Þ ¼ ðwnew ; wnew L R Þ−0:5∑ðΔwL ; ΔwR Þ þ

Linear plasticity rule, divisive normalization

1

wL+ΔwL

the postsynaptic neuron. In all simulations, the initial point was (wL,wR)¼(0.5,0.5) and the learning rate α was 0.01, except for Figs. 2 and 3A where α¼0.4 for illustration purposes. Weights were constrained to positive values only. For the constrained-connectivity model, weight were further constrained to the range [0,1]. Normalization in the “constrained-connectivity” model: At each time step, after updating the connectivity weights, these were normalized to maintain a fixed-sum

197

0.005

0

−0.005

3. Mathematical Derivations −0.01

3.1. Mathematical Derivation 1: Changes in connectivity-weights following a pair of left- and right-eye activations with a linear Hebbian plasticity rule and divisive normalization Initial weights: wL and wR. Step I: Activation of left-eye: ( I wL ¼ wL þ ΔwL ¼ wL þ αxL y ¼ wL þ α1ð1wL þ 0wR Þ ¼ wL þ αwL wIR ¼ wR þ ΔwR ¼ wR Step II: Divisive normalization: 8 > < wIIL ¼ > : wIIR ¼

wIL ∑wI

¼

wL þαwL wR þwL þαwL

wIR

¼

wR 1þαwL

∑wI

¼

wL þαwL 1þαwL

1þα ¼ wL 1þαw L

0

0.25

0.5

0.75

1

wL Fig. 2. Dynamical system representation of the model and connectivity change following a pair of eye activations. (A) Model dynamics for a linear Hebbian plasticity rule and divisive normalization. The updated value of wL (i.d., wL+ΔwL) after activation of either left- or right-eye-input was calculated analytically as a function of wL (ocher/ blue curves respectively; see Mathematical Derivation 3.1). The black trace illustrates the progression of wL during four time steps (order of input activation: L,L,L,R). At each time step, the value of wL (abscissa) is marked along the diagonal (dotted line); from this point, a vertical line is drawn to the curve corresponding to the activated input (up/down for left/right-eye input, respectively); and finally a horizontal line is drawn to the diagonal, to “update” the value of wL along the abscissa. (B) Simulation of 50 time steps, where at each time step one of the two inputs was selected randomly and activated (input activation starting at L,R,R,R,L,R,L,L…). In this example, wL fluctuates back and forth around the initial value of 0.5 (cf. Fig. 1B). (C) Net change in ΔwL after a pair of left- and right-eye activations as function of wL. The zero net change in connectivity following a pair of eye activations indicates a marginally stable dynamics, in which the connectivity weights do not diverge over time, and lack of binocular competition. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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M. Furman / Neurocomputing 121 (2013) 195–206

Step III: Activation of right-eye (cf. step I) 8 II 1þα < wIII L ¼ wL ¼ wL 1þαwL II II : wIII R ¼ wR þ ΔwR ¼

wR 1þαwL

þ αwIIR ¼

wR 1þαwL

wR 1þα þ α 1þαw ¼ wR 1þαw L L

Step IV: Divisive normalization: III ΣwIII ¼ wIII L þ wR ¼ ðwL þ wR Þ

8 < wIV ¼ L : wIV R

¼

wIII L ∑wIII

1þα 1þα ¼ 1 þ αwL 1 þ αwL

To reflect the first curve with respect to the diagonal, the roles of x and y can be switched as follows: x¼y

1þα 1 þ αy 1þα 1 ¼ 1 þ αy y x y

þα ¼

1þα x -y¼ x 1 þ αð1−xÞ

As can be seen, the result is identical to the second curve, thus demonstrating their symmetry with respect to the diagonal.

1þα 1þαwL ¼ wL 1þαw ¼ wL L 1þα

1−wIV L

¼ 1−wL ¼ wR

4. Results 4.1. Binocular competition through constrained connectivity: model formulation

3.2. Mathematical Derivation 2: Changes in connectivity weights following a pair of left- and right-eye activations with a linear Hebbian plasticity rule and subtractive normalization Initial weights: wL and wR. Step I: Activation of left-eye (cf. Derivation I, step I) ( I wL ¼ wL þ αwL wIR ¼ wR Step II: Subtractive normalization: ( II wL ¼ wIL −0:5ðΔwL þ ΔwR Þ ¼ wIL −0:5αwL ¼ wL þ 0:5αwL wIIR ¼ wIR −0:5ðΔwL þ ΔwR Þ ¼ wIR −0:5αwL ¼ wR −0:5αwL Step III: Activation of right-eye (cf. step I) (

II wIII L ¼ wL ¼ wL þ 0:5αwL II II II wIII R ¼ wR þ ΔwR ¼ wR −0:5αwL þ αwR ¼ wR −0:5αwL þ αðwR −0:5αwL Þ

Step IV: Subtractive normalization: (

III III III wIV L ¼ wL −0:5ðΔwL þ ΔwR Þ ¼ wL þ 0:5αwL −0:5αðwR −0:5αwL Þ III III III wIV ¼ w −0:5ðΔw þ Δw R R L R Þ ¼ wR −0:5αwL þ αðwR −0:5αwL Þ−0:5αðwR −0:5αwL Þ

The overall change in wL can be further simplified and expressed as a function of wL alone (shown in Fig. 3B) 2 wIV L −wL ¼ 0:5αwL −0:5αð1−wL −0:5αwL Þ ¼ α½wL −0:5 þ 0:25α wL

3.3. Mathematical Derivation 3: Testing for symmetry with respect to the main diagonal in Fig. 2A B For a linear Hebbian plasticity rule and divisive normalization, wL following left-eye activation is (see Mathematical Derivation 1 step II): wL

1þα 1 þ αwL Similarly, wL following right-eye activation is

wL

1 1 ¼ wL 1 þ αwR 1 þ αð1−wL Þ Accordingly, the two curves displayed in Fig. 2A B are

y¼x

1þα 1 ;y¼x 1 þ αx 1 þ αð1−xÞ

In visual areas of the mammalian brain, such as the LGN and SC, inputs from the two eyes are mixed early during development, and then segregate into eye-specific domains. Each of these eye-specific domains eventually receives input predominantly from one of the two eyes. This process of eye-specific segregation can be formalized as the evolution of two variables, wL and wR, representing connectivity weights from the left- and right-eye, respectively, onto a target neuron (Fig. 1A). As an approximation, one may assume that initially the two weights are equal. If eye-segregation proceeds successfully, one of the two weights increases whereas the other approaches zero; otherwise, the two weights remain close to 0.5. The constrained-connectivity model of binocular competition is based on two principles: (1) strengthening, at each time step, the connectivity from an active (or stimulated) eye; (2) competition between the two eyes, implemented through weight normalization, which maintains the sum of the two weights fixed over time. The normalization constraint ensures that if at a given moment one eye is active whereas the other is not, the connection of the non-active eye to the target is weakened to preserve the sum of the two weights fixed. In a concrete formulation of the model, one would need to specify (1) a plasticity rule: the specific activity-based rule for weight modification; and (2) a weight normalization rule: the algorithm for maintaining the fixed sum-weights. The plasticity and normalization rules will determine the overall dynamical behavior of the model. For a specific simulation, one would need to further indicate the initial conditions (wL ð0Þ; wR ð0Þ), and the temporal pattern of activation of the two eyes. Simulations in this paper assume that over time, the overall activity of the two eyes is balanced, as one would expect biologically under normal conditions. Two commonly used forms of weight normalization, which are the focus of the present paper as well, are divisive and subtractive. In divisive normalization (sometimes referred to as multiplicative), at each time step, after adjusting the weights according to the weight-modification rule, the two weights are divided by their sum. In other words, each of the two weights is adjusted in proportion to its magnitude (e.g., [20,26,27]). In contrast, in the subtractive normalization rule, half of the total weight increment is subtracted from the two connection weights; that is: the two weights are adjusted by the same amount, independently from their magnitude (e.g., [19,22]). In principle, other forms of weight normalization, such as fixed sum-of-squared-weights, may be considered as well [33,34]. The constrained-connectivity model is the focus of Sections 4.2–4.4. Section 4.5 introduces a new, modified model, based on recent physiological data. This model posits that the total connectivity weight is not fixed over time but rather increases in response to the competition between the two eyes. The formalism of this “growthpromoting” model is along the same lines as described above, but the

M. Furman / Neurocomputing 121 (2013) 195–206

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sum-of-weights is increased by a certain fraction β whenever one of the inputs is active (see Methods). 4.2. Normalization constraint affects model dynamics through the net change in connectivity following a pair of left- and right-eye activations How, specifically, does competition operate in the constrainedconnectivity model? The normalization rule ensures that at each time step one eye is strengthened whereas the other is weakened. Is this strengthening/weakening coupling sufficient by itself to generate binocular competition? The answer is not necessarily. The model may, or may-not display competitive dynamics, depending on the specific form of normalization constraint, even when one of the eyes is strengthened and the other is weakened at each time step. As a starting point, Fig. 1B shows a simulation of the temporal evolution of wL and wR in a model with a basic, linear Hebbian plasticity rule, Δwi ¼ αxi y complemented by divisive normalization, and a linear transfer function, y ¼ xL wL þ xR wR At each time step, one of the two eyes was selected at random and activated. Over time, wL and wR fluctuate around 0.5 but remain around this value. A fundamentally different picture emerges by substituting divisive by subtractive normalization (Fig. 1C). Here one of the weights increases towards 1 whereas the other goes to 0. This difference between divisive and subtractive normalization rules was previously reported [19,21,22]. It was analyzed, for a more general context of multiple input and output neurons, using an eigenvector decomposition of the input correlation matrix. Here, I propose an additional approach for explaining these differences, taking advantage of the simplified context of binocular competition prior to eye opening, which can be modeled by alternating activation of the two eyes. As mentioned, the change in connectivity after activation of a single-eye is not particularly informative for clarifying the behavior of the model. Alternatively, one could ask: What is the change in connectivity weights after a pair of left- and right-eye activations? As shown next, the change in connectivity after a pair of activations explains the presence or absence of competitive dynamics in the model. For a linear Hebbian plasticity rule and divisive normalization, the net change in wL and wR after left- and right-eye activations is, interestingly, precisely zero for all values of wL (see Mathematical Derivation 3.1 and Fig. 2C). In other words, when activity of the two eyes is balanced over time, fluctuations in connectivity weights cancel out on average over time. One way to represent this process is using a diagram borrowed from the field of dynamical systems analysis and adjusted to the context of binocular competition. The proposed diagram makes use of the fact that at any time step wR ¼ 1 −wL. This allows focusing on the dynamics of just one of the two weights, e.g. wL. Fig. 2A illustrates this idea for the case of linear Hebbian plasticity rule and divisive normalization. The ocher and blue curves display the updated values of wL (including weight normalization) after a left- or right-eye activation, respectively, as a function of wL. The dynamics of wL can be traced by drawing, at each time step a vertical line towards the curve representing the active eye, and from there a horizontal line to the diagonal wL ¼wL. In this way, the updated value of wL after each iteration is re-transferred to the abscissa. The lack of binocular competition in this specific formulation of the model can be derived from this diagram. Because the ocher and blue curves are symmetric to the diagonal (see Mathematical Derivation 3.3), wL would fluctuate over time but its average drift would be zero (Fig. 2B).

Fig. 3. Competitive dynamics under a linear plasticity rule and subtractive normalization. (A) Dynamical-system representation of the model (same formalism as in Fig. 2A). Note that in contrast to divisive normalization (Fig. 2A), here ΔwL is larger for left- than for right-eye activation in the range 0.5 o wL o 1. Consequently, after a pair of left- and right-eye activations, the net change in ΔwL is positive, as shown in panel B.

What then happens when divisive normalization is substituted by subtractive normalization? Now the changes in wL after a leftor right-eye activations do not cancel out (Fig. 3A). Consequently, even when the activity of the two eyes is balanced over time, the weights drift towards one of the stable points (0,1) or (1,0). Fig. 3B shows the change in wL after a pair of left- and right-eye activations (Mathematical Derivation 3.2). Interestingly, now ΔwL 40 for wL 40.5, whereas ΔwL o0 for wL o0.5. In other words, after a pair of left- and right-eye activations, the larger of the two weights increases, whereas the other decreases. This is the signature of a “positive feedback” competitive dynamics, sometimes referred to as “rich get richer”. In this scenario, differences between wL and wR are gradually amplified. One advantage of the analysis presented above is that it can be easily generalized to various forms of plasticity rule, normalization constraint and transfer function. Even in cases where the change ΔwL resulting from a pair of left- and right-eye activations cannot be calculated analytically, a semi-analytic approach can be employed where ΔwL is calculated numerically for a large set of wL values, and

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M. Furman / Neurocomputing 121 (2013) 195–206

Fig. 4. Competitive dynamics under divisive normalization and non-linear plasticity rules. (A) Eye-segregation score as a function of time from 20 simulations of the plasticity rule Δwi ¼ αxiwiy, divisive normalization and a linear transfer function. In each simulation, wL and wR were initially equal (eye-segregation score equals zero). The gradual increase in the eye-segregation score indicates divergence of the two weights and therefore reflects competitive dynamics between the two eyes. (B) ΔwL following a pair of left- and right-eye activations as a function of wL, calculated numerically at 0.005 steps of wL. (C) and (D) Same as A and B for a sigmoidal transfer function (see text). Note the different time scales in A vs. C.

then plotted as a function of wL, to generate the sort of diagram as in Figs. 2C and 3B and elucidate the model's dynamics. 4.3. Sensitivity to plasticity rule formulation The constrained-connectivity model is sensitive not only the specific normalization rule being used but also to the details of the plasticity rule [21]. I will focus on divisive normalization, as its relevance to the biological system is more firmly established than subtractive normalization [24]. Elliott [21] examined a family of plasticity rules, Δwi ¼ ρðxi Þsðwi ÞΦðx⋅wÞ and demonstrated that the following conditions guarantee eyespecific segregation (in the context of the binocular competition model) under a divisive normalization constraint: ðiÞ sð0Þ ¼ 0; ðiiÞ sðsÞ≥As; and ðiiiÞΦ′ðsÞρ′ðsÞ≥0 (the formalism of Elliott [21] contains two additional functions, f and g; also, the original nomenclature was modified for clarity). The basic linear Hebbian plasticity rule violates the first condition, because in this rule s(s)¼ 1. However, various modified Hebbian plasticity rules do respect the three conditions specified above, and therefore support eye-specific segregation. One of the simplest modified rules is obtained by substituting s(s)¼1 by s(s) ¼s [26,27,35], that is: Δwi ¼ αxi wi y Fig. 4A shows simulations of the model using this plasticity rule and divisive normalization. Fig. 4B shows ΔwL following a pair of left- and right-eye activations in this formulation of the model. In

contrast to the basic linear Hebbian rule, here ΔwL is positive for wL 4 0.5, reflecting competitive dynamics between the two eyes. Another generalization of the basic linear model is non-linearity in the transfer function of the postsynaptic neuron; a sigmoidal transfer function, for instance, can be implemented as follows, y ¼ ΦðxL wL þ xR wR Þ ΦðtÞ ¼

1 ; k1 4 0; k2 40 1 þ k1 e−k2 t

This function is monotonically increasing, and in conjunction with the plasticity rule Δwi ¼αxiwiy for instance, preserves competitive dynamics in the model. This is illustrated in Fig. 4C and D for k1 ¼5, k2 ¼ 10. To summarize Section 4.3, some, but not all plasticity rules generate competitive dynamics in the constrained-connectivity model under divisive weight normalization. The presence or absence of binocular competition is reflected in the net connectivity change following a pair of left- and right-eye activation: competitive dynamics requires a positive change in connectivity for the larger of the two weights. 4.4. Does the temporal pattern of eye-activation matter? Until recently, the temporal pattern (over long time scales) of spontaneous retinal activity prior to eye opening was unclear, because most studies of retinal activity at this developmental stage were based on isolated retinal preparations. Recently, in vivo experiments have provided direct evidence for retinal waves prior to eye opening, including specific data on the distribution of interwave intervals [7].

M. Furman / Neurocomputing 121 (2013) 195–206

Table 1 Examples of input activity patterns with varying degrees of temporal correlation. γ

Example of input pattern

γ ¼0a γ ¼0.25 γ ¼0.50 γ ¼0.75 γ ¼1.0

LRLRLRLRLRLRLRLRLRLR…b LRRLRLRLRLRLRRRRLRLR… RLLLRLLRLLRRLLLLLRRL… RRRRRLLRRRRLLRRRLLLL… RRRRRRRRRRRRRRRRRRRR…

a γ indicates the probability that the active eye at a given type step is the same as the eye activated during the previous time step (see text). b L/R indicate activation of the left- and righteye inputs, respectively.

Fig. 5. The temporal pattern of input activity affects eye-segregation rate. (A) Different input activity patterns were generated by varying γ, a parameter determining the probability that the active eye at a given type step is the same as the eye activated during the previous time step (see Table 1 for examples of input activation pattern). In all cases, activity of the two eyes is fully decorrelated, but the distribution of intervals between consecutive activations of the same eye varies as a function of γ (see Fig. 6). Each trace shows the eye-segregation score as a function of time averaged over 50 simulations. (B). The time required to reach a 0.8 eyesegregation threshold, as a function of γ (see Methods). Gray line shows linear fit at point of minimal slope (0.51, slope −704.7). The theoretical value of γ at the minimal slope point corresponds, approximately, to γ value fitting the experimental distribution of inter-wave intervals (see Fig. 6 and text).

From the theoretical perspective, the question arises whether the temporal pattern of input activation affects the speed of eye segregation in those cases where competitive dynamics does take place in the model. Addressing this question analytically is difficult,

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because as connectivity weights progress away from the stable point (0.5,0.5), the step size of the changes in wL and wR vary as a function of the weights themselves. Computationally, however, different patterns of input activation can be generated to test their effect on segregation time. To generate input patterns of varying temporal characteristics, one can define a parameter γ determining the probability that the active eye at a given type step is the same as the eye activated during the previous time step. Specifically, γ¼0 generates a pattern where the activated eye alternates systematically; γ¼0.5, in contrast, generates an input pattern “without memory”, as in previous sections, where at each time step the active eye is selected at random; and finally, 0oγo0.5 and 0.5oγo1.0 generate patterns where the probability of consecutively activating the same eye is smaller/larger than chance, respectively (Table 1). For simplicity, I will focus on divisive normalization and the plasticity rule Δwi ¼ αxi wi y with a linear transfer function y¼ xLwL+xRwR. It should be noted that for all γ values, except for γ¼1, activity of the two eyes is balanced over time (i.e., the overall probability of activating the left-eye, for instance, is 0.5). Thus, γ controls input correlations at short time scales, without generating imbalance between the overall activities of the two eyes. It should also be noted that under all these conditions activity of two eyes is fully de-correlated, as only one eye is active at a time and as the plasticity rule operates at each time step separately, without “memory”. As shown in Fig. 5, γ has a dramatic effect on the rate of eyesegregation in the model. Specifically, increasing γ (i.e., increasing the probability of same-eye activation in consecutive time steps), accelerates eye-segregation. From the physiological perspective, these simulations indicate that the sequence of retinal waves among the two eyes, and not only the spatio-temporal structure within each retinal wave, may impact formation of binocular maps in the brain. To compare the simulated patterns of input activations to the experimental data of Ackman et al. [7], one can calculated the distribution of inter-wave intervals for each value of γ (with interwave interval defined as the number of time steps between consecutive activations of the same eye). The resulting distributions for γ ¼ 0:25; 0:5; 0:75 are shown in Fig. 6, along with the coefficient of variation (CV), the ratio between standard deviation to the mean, for each distribution. The experimental CV value is approximately 0.8 (mean inter-wave interval of 40.5sec and standard deviation 32.4 s). This experimental value is adequately represented by the simulated distribution of γ¼0.5 (CV ¼0.714). Best fit to the experimental value is obtained at γ ¼0.566. Can the model suggest an adaptive role for the experimentally observed temporal pattern of retinal waves? Fig. 5B shows the average simulated time required to reach an eye-segregation score of 0.8 as a function of γ (where γ is the probability that the active eye at a given type step is the same as the eye activated during the previous time step; thus, γ determines the temporal pattern of input activation). Interestingly, the minimal slope of this curve corresponds to γ ¼0.51, close to the value representing the experimentally observed distribution of inter-wave intervals. In other words, the simulated input pattern that matches the experimental data corresponds to the parameter region where eye-segregation is minimally sensitive to γ (see Discussion). 4.5. A “growth-promoting” model of binocular competition The constrained-connectivity model, discussed in previous sections, posits that inputs from the two eyes compete under a fixed-sum constraint. After eye opening, homeostatic synaptic scaling, as observed empirically in the visual system [23,24], offers a potential biological implementation of the normalization constraint. However, binocular competition exists also prior to eye

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Fig. 6. Relationship between the parameter γ and the distribution of “inter-wave intervals” in the model. (A)–(C). For comparison with experimental data, “inter-wave interval” was defined as the time difference between consecutive activations of the same eye. Panels A–C shows the distribution of inter-wave intervals for γ ¼ 0:25; 0:50; 0:75, respectively. Also indicated are the coefficients of variation (CV, ratio between standard deviation to the mean) for each distribution. The two additional γ values shown in Fig. 5A, γ ¼0 and γ¼ 1 yield fixed inter-wave intervals of 2 and 1 time steps, respectively (CV ¼ 0). γ ¼ 0.566 provides best fit to the experimental value of CV ¼ 0.8 [7].

opening. In contrast to post-eye-opening development, binocular competition before eye opening is mediated by spontaneous activity in the developing retinas. Recent patch-clamp recordings from neonatal mice prior to eye opening have shown that in fact, binocular competition prior to eye opening enhances synaptic connections between retinal ganglion cells and their target neurons, instead of operating under a fixed-sum constrained as after eye opening [32]. In other words, these findings suggest that the normalization-constraint is “released” before eye opening, and instead, binocular competition results in a gradual increase in total connectivity. To model “growth-promoting” binocular competition, one could modify the classical constrained-connectivity model by increasing the total connectivity wL þ wR by a factor β at each time step. This can be implemented using either a divisive or subtractive weight adjustment, comparable to the divisive/subtractive normalization rules in the constrained-connectivity model. In divisive weightadjustment the connectivity weights are scaled to achieve the desired growth in total connectivity; and in subtractive weightadjustment, a factor is subtracted from each of the connectivity weights, again to achieve the desired change in summed connectivity (see Methods). How does the growth-promoting model compare to the constrained-connectivity model in generating binocular competition? In contrast to the constrained-connectivity model, the growth-promoting model displays competitive dynamics under a basic linear Hebbian plasticity rule and divisive weight adjustment (Fig. 7A). After a pair of left- and right-eye activations, the change in connectivity is larger for the larger of the two weights. This indicates a positive-feedback loop in which differences in connectivity are amplified. Indeed, simulations show that the two connection weights diverge in this model leading eventually to high eye-segregation scores in all simulations (Fig. 7A, middle and right panels). If so, unlike the constrained-connectivity model, the growthenhancing model displays competitive dynamics under a basic linear Hebbian plasticity rule and divisive weight adjustment. However, the biological relevance of this specific formulation of the model is limited, particularly because physiologically, the absolute increase in connectivity due to binocular competition prior to eye opening is in the range of 20–50% [32], whereas in this specific formulation of the growth-enhancing model there is hundreds to thousands percent increase in connectivity by the end of the simulations (Fig. 7A). In contrast, other formulations of the model, for instance linear Hebbian rule with subtractive weight adjustment (Fig. 7B) or a modified plasticity rule Δwi ¼αxiwiy with divisive weight adjustment (Fig. 7C), display competitive dynamics and biologically-realistic increases in connectivity. At closer inspection, there are number of major differences between (i) the growth-enhancing model with basic linear

Hebbian plasticity rule and divisive weight adjustment (Fig. 7A) and (ii) the growth-promoting model with either linear Hebbian rule and subtractive weight adjustment (Fig. 7B) or the modified plasticity rule Δwi ¼αxiwiy and divisive weight adjustment (Fig. 7B and C). Specifically, with basic linear Hebbian plasticity rule and divisive weight adjustment, (1) the change in ΔwL −ΔwR following a pair of input activations is substantially smaller than in the two other formulations for the same parameters; (2) accordingly, eyesegregation proceeds at a rate roughly two orders of magnitudes slower; (3) the relative increase in connectivity by the end of simulations is large (hundreds to thousands percent increase in connectivity); and (4) in the model with basic linear Hebbian plasticity rule and divisive weight adjustment (Fig. 7A), both connection weights increase (in contrast to only one of the two weights in Fig. 7B and C), although the magnitudes of the two weights diverge, leading eventually to a high eye-segregation score (Fig. 7A, right panel). Finally, how does the temporal pattern of input activity affect segregation in the growth-promoting model? Fig. 8 shows the time required to reach a 0.8 eye-segregation threshold in the growth-promoting model as a function of γ (see also Section 4.4). Comparison to the constrained-connectivity model (Fig. 5B) shows reduced sensitivity of the growth-promoting model to γ in terms of the time required to reach eye-segregation. This suggests that the growth-promoting model is more robust to variations in the temporal structure of input activation.

5. Discussion In the developing mammalian visual system, inputs from the two eyes compete during development for innervation of target neurons. A common model for binocular competition is the constrainedconnectivity model, in which connections from the two eyes compete through a fixed-sum constraint. The constrained-connectivity model has been previously analyzed using a correlation-based approach, particularly suitable for modeling visual development after eyeopening. Prior to eye opening, binocular competition is mediated by spontaneous waves of retinal activity, resulting in largely decorrelated activity between the two eyes. The first goal of this paper was to propose a simplified analysis of the constrained-connectivity model for visual development prior to eye-opening. It is shown here that the presence or absence of competitive dynamics in the model can be examined and explained based on the change in connectivity following a pair of left- and right-eye activations. When this net change is positive for the larger of the two weights, the model generates a positive-feedback loop that amplifies differences in connectivity between the two eyes. Together with a dynamical system visualization of the model, these findings offer an explanation

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Fig. 7. Competitive dynamics in a “growth-promoting” model of binocular development. To model “growth-promoting” binocular development as observed experimentally prior to eye opening [32], the fixed-sum constraint on connectivity was replaced by a fractional increase in connectivity at each time step (see Methods). (A) Analysis and simulations using a basic Hebbian plasticity rule with divisive weight adjustment. Left: Difference in weight changes ΔwL−ΔwR following a pair of left- and right-eye activations as a function of wL. Since in this model weights are not constrained to wL+wR ¼ 1, the difference in weight changes is used instead of ΔwL (cf. Figs. 2C, 3B and 4B and D). For wL 4 0.5, ΔwL–ΔwR is positive, indicating larger strengthening of the stronger of the two weight. Thus, this model, in contrast to the constrained-connectivity model with basic Hebbian plasticity and divisive normalization, displays competitive dynamics. This is confirmed by direct simulations (middle and right panels): in all simulations, eye segregation score reaches a 0.95 threshold value, and the two weights diverge. However, the time scales of eye-segregation in this version of the model are slow (see text). The shown examples of wL, wR time course (right) were truncated at time step 10,000 for clarity; in most simulations in eye-segregation was reached at a later time point. (B) and (C). Analysis and simulations for a basic Hebbian plasticity rule with subtractive weight adjustment, and a modified Hebbian rule with divisive weightadjustment, respectively. All simulations were terminated when eye-segregation reached a threshold value of 0.95.

for the model's sensitivity to formulation details. Secondly, the paper proposes a new, modified model, based on recent physiological data, in which competition between the two eyes enhances connectivity instead of operating under a fixed-sum constraint. It is shown that this model is more robust to formulation details, suggesting that this “growth-promoting” dynamics potentially provides adaptive advantage during early stages of visual development. The constrained connectivity model for binocular competition has been previously analyzed using eigenvector decomposition and stability analysis of the fixed points of the dynamics [19,21]. This approach is particularly fruitful for modeling the formation of ocular dominance (OD) columns/stripes in the visual cortex in response to visual experience. These and other models (e.g. [5,36,37]) demonstrated a crucial role for the degree of interocular correlation in the development of the spatial properties of OD columns. Network models based on Hebbian plasticity have also been applied the formation of binocular maps in subcortical visual areas prior to eye opening [13,35,38–41]. Network level

modeling is necessary for addressing the concurrent formation of retinotopic maps, binocular maps, and other processes such as ON/OFF segregation (also see [42]), but at the same time somewhat complicates the interpretation of how precisely inputs from the two eyes compete and segregate. By focusing on a simplified architecture consisting of just two inputs converging onto a single target cell, this paper proposes an additional approach for understanding binocular competition, based on a dynamical-system diagram to visualize changes in connectivity following input activation, and analytic or numeric calculation of the change in connectivity weights following a pair of left- and right-eye activations. An advantage of the proposed analysis is that it can be easily applied to modified versions of the constrained-connectivity model that are harder to tackle using the eigenvector decomposition approach, including the growth-promoting model discussed here. The sensitivity of the constrained-connectivity model to the type of normalization constrained (Section 4.2) and plasticity rule

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Fig. 8. Eye-segregation in the growth-promoting model as a function of the temporal pattern of input activation. Time required to reach a 0.8 eye-segregation threshold in the growth-promoting model, as a function of γ that controls the temporal statistics of input activation (cf. Fig. 5B and see Methods for details). Note the smaller dynamic range, 1930 time steps (½min; max ¼ ½301; 2231) in this model compared to 2496 time steps (½min; max ¼ ½311; 2807) in the constrained-connectivity model (Fig. 5B), as well as the smaller slope (−472.8 minimal slope compared to −704.7 in the constrained-connectivity model). Together, these two characteristics indicate reduced sensitivity of the growth-promoting model to changes in γ.

(Section 4.3) was examined in [19,21,38], and the analysis presented here complements these studies. It is shown here that a linear Hebbian plasticity rule together with divisive normalization leads to a unique condition where weight changes following leftand right-eye activation cancel out, generating a marginally stable, non-competitive dynamics (Fig. 2). By contrast, substituting the divisive by subtractive normalization (Fig. 3), or substituting the linear Hebbian rule by certain non-linear rules (Fig. 4), leads to competitive dynamics. Under these conditions, a pair of left- and right-eye input activations strengthens the larger of the two input weights. The conditions examined in Sections 4.2 and 4.3 represent a subset of the possible combinations of plasticity and normalization rules. One may wonder how the constrained-connectivity model would behave under other conditions. Many combinations of plasticity and normalization rules lead to dynamics that is quite hard to approach analytically. The semi-analytic approach outlined in Section 4.2 and Figs. 2–4 can be used in such cases to test whether the model leads to competitive dynamics or not. Since exhausting all possible combinations of plasticity and normalization rules is obviously impractical, a productive approach would be to focus on model formulations that are most physiologically plausible (based on ongoing advances in experimental work), and apply either a simplified yet more limited analysis as proposed here, or, whenever possible, a more sophisticated and in-depth mathematical analysis. It is important to note that weight normalization is not the only way to formalize a constraint on total connectivity. One alternative is competition for limited neurotrophic resources [43–46]. Competition for neurotrophic factors is biologically plausible (e.g. [47]), although the precise mechanisms of neurtrophic-based competition during mammalian visual map development are unclear. Interestingly, Elliott and Shadbolt [27] showed that the dynamics underlying competition for neurotrophic factor, as implemented in their previous study, is equivalent to a combination of a non-linear Hebbian rule and divisive weight normalization. More generally, various additional modeling approaches for axonal competition have been proposed [48–53]; for reviews, see Van Ooyen [54] and Swindale [55].

When competitive dynamics is manifested in the model, through appropriate choice of plasticity rules and normalization constraints, the rate of eye-segregation depends on the temporal pattern of input activity (Figs. 5 and 8). It is important to note that in this model the learning rule operates at each time step independently, without memory or temporal delays. Thus, changing the temporal pattern of activity does not affect the correlation between the two eyes, and therefore it is not clear a priori whether and how these temporal patterns should affect eye-segregation. Simulations show that the time required for reaching eye-segregation strongly depends on the temporal structure of input activity (Fig. 5). Recently, the temporal structure of retinal waves over long time scales was experimentally characterized in vivo for the first time [7]. Thus, using the distribution of inter-wave intervals, theory and experiments can be directly related (Fig. 6). Interestingly, the conditions matching the experimental data (γ¼0.566) closely match the parameter region where the time required for eye-segregation is minimally sensitivity to γ. In other words, the model suggests that the distribution of interwave intervals as observed biologically may contribute to developmental stability by minimizing sensitivity to perturbations in the temporal statistics of retinal waves. The interval between consecutive retinal waves is thought to be dominated by refractoriness in RGCs [56], and any environmental or genetic factor that interferes with the cellular or network mechanisms of this refractoriness is likely to alter the temporal pattern of retinal waves. Recently, the idea of competition under constrained-connectivity was tested experimentally for the developmental period before to eye opening. It was found that in contrast to the predictions of this model, binocular competition prior to eye opening enhances synaptic and dendritic growth rather than operating under a fixed-sum constraint on connectivity [32]. This suggests that the classical constrained-connectivity model, which is well-established for binocular development after eye opening, appears to be less adequate for earlier stages in development, prior to eye opening. Here I propose a relatively simple modification of the constrained-connectivity model that allows (1) modeling binocular competition that enhances connectivity, as observed experimentally; (2) direct comparison between the more traditional constrained-connectivity and the proposed “growth-promoting” model. The proposed modification involves a single additional parameter, β, which indicates a fractional increment in total connectivity at each time step. Divisive and subtractive normalization rules of the constrained-connectivity model are generalized to divisive and subtractive weight-adjustment rules, respectively, that instead of maintaining fixed-sum connectivity, increase the total connectivity by a fraction β. Analysis and simulation (Fig. 7) show that in contrast to the constrained-connectivity model, the growth-promoting model generates competitive dynamics under divisive weight-adjustment and a basic linear Hebbian plasticity rule, unlike the constrainedconnectivity model that fails to generate competition under similar conditions. It is important to note, however, that quantitatively, for realistic amounts of connectivity growth, the growth-promoting model with divisive weight-adjustment and basic linear Hebbian plasticity exhibits poor eye-segregation. Thus, theoretically, the growth-promoting model displays binocular competition under broader conditions, but only some formulations bear direct relevance to the physiological reality. Under formulations where both the “growth” and “non-growth” models produce competitive dynamics, both these models produce qualitatively similar behavior, at least when the growthenhancing parameter β is chosen to produce biologically-realistic total increase in connectivity, that is, in the range of 20–50% (compare Figs. 3–7; see also Section 4.5). However, it is possible that biologically, the growth-promoting mode of axonal competition supports developmental processes beyond binocular competition, such as stabilizing synapse formation or supporting intra-

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cellular signaling pathways. Testing these issues theoretically would require a more biophysically-detailed computational model of axonal competition. The growth-promoting model shows reduced sensitivity to changes in the temporal pattern of input activity compared to the constrained-connectivity model (Figs. 8 vs. 5). Generally, the temporal pattern of input activity determines the specific sequence of weight increases and decreases over time. In the growth-promoting model the positive-feedback of strengthening the larger of the two inputs is accentuated by the fact that not only the individual weights but also the total connectivity weight varies over time. This overall increase in connectivity, irrespective of the specific changes at each time-step, overrides to some extent the role of the temporal pattern of input activity and thereby reduces the sensitivity of the model to it. Together, these simulations results indicate that the growthpromoting model represents a somewhat more robust form of binocular competition compared to constrained-connectivity, and may therefore fit better the biological context of the developmental period before eye opening. The biophysical mechanisms of growthpromoting binocular competition are unknown, and are likely to involve axon–axon interactions [57,58] in addition to axon–target interactions, which underlie the more traditional forms of Hebbianlike mechanisms. Future progress in understanding axonal competition during brain development would allow detailed and more biologically-realistic modeling of this process.

Acknowledgments I gratefully acknowledge Dr. Michael Crair and members of the Crair laboratory at Yale University for valuable discussions and feedback on this work. I also would like to thank the two anonymous reviewers for their helpful and constructive feedback.

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Moran Furman received his Ph.D. in Biomedical Engineering from the Technion, Israel Institute of Technology in 2004. He is currently an Associate Research Scientist in the Department of Neurology at Yale University School of Medicine, working on the neural basis of sensory processing during normal and pathological brain states. He has authored/co-authored computational and experimental articles in numerous neuroscience journals including Neuron and The Journal of Neurophysiology. His research interests encompass the function and development of the visual system, mechanisms of loss of consciousness during epileptic seizures, neural-network modeling of cognitive functions, and more.