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How does non-random spontaneous activity contribute to brain development? Jean-Philippe Thivierge ∗ Department of Psychological and Brain Sciences, Indiana University, 1101 East Tenth Street, Bloomington, IN 47405, United States

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Article history: Received 15 January 2008 Received in revised form 17 July 2008 Accepted 1 January 2009 Keywords: Retinocollicular projections Development Spike-timing-dependent plasticity Spontaneous activity Topographic map

a b s t r a c t Highly non-random forms of spontaneous activity are proposed to play an instrumental role in the early development of the visual system. However, both the fundamental properties of spontaneous activity required to drive map formation, as well as the exact role of this information remain largely unknown. Here, a realistic computational model of spontaneous retinal waves is employed to demonstrate that both the amplitude and frequency of waves may play determining roles in retinocollicular map formation. Furthermore, results obtained with different learning rules show that spike precision in the order of milliseconds may be instrumental to neural development: a rule based on precise spike interactions (spike-timing-dependent plasticity) reduced the density of aberrant projections to the SC to a markedly greater extent than a rule based on interactions at much broader time-scale (correlation-based plasticity). Taken together, these results argue for an important role of spontaneous yet highly non-random activity, along with temporally precise learning rules, in the formation of neural circuits. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In several models of neural and cognitive processes, the brain is conceptualized as an input–output system mainly driven by its interactions with the external world (Llinas, 2001). However, in recent years, this view has been challenged by evidence that intrinsic fluctuations in neural activity (not directly linked to environmental input) influence not only the interactions between neural regions (Honey, Kotter, Breaksprear, & Sporns, 2007), but ultimately contribute to both behavioral (Fox, Snyder, Vincent, & Raichle, 2007) and cognitive responses, including memory (Pessoa, Gutierrez, Bandettini, & Ungerleider, 2002) and attention (Sapir, D’avossa, Mcavoy, Shulman, & Corbetta, 2005). Given that the majority of brain activity is spontaneous, and that spontaneous forms of neural activity have been reported throughout the central nervous systems of a variety of vertebrate species (Feller, 1999), it is not difficult to conceive of this form of activity as having an influence on a wide array of neural processes. What influence might spontaneous activity exert on the development of neural circuits? Given its highly non-random nature (Cossart, Aronov, & Yuste, 2003; Ikegaya et al., 2004; Mao, Hamzei-Sichani, Aronov, Froemke, & Yuste, 2001), it seems unlikely that spontaneous activity merely adds noise to developmental processes. Rather, the development of brain circuits appears to directly depend on the structure of spontaneous activity (McLaughlin, Torborg, Feller, & O’Leary, 2003); in other words,

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non-random aspects of endogenous activity may provide information on how neural circuits should be wired through the course of development. For instance, the development of a topographic map that relays visual information from retinal ganglion cells (RGCs) to the superior colliculus (SC) (McLaughlin & O’Leary, 2005; Thivierge & Marcus, 2007) depends on patterned activity. In an initial stage of development, molecular guidance cues (e.g., Ephs and ephrins) (Thivierge & Marcus, 2007) give rise to a rough initial map (Fig. 1a). Then, during a so-called ‘‘pre-critical’’ period, spontaneous activity strengthens coactive synapses, and prunes out extranumerary projections. This activity takes the form of large waves of action potentials that propagate slowly across the retina. As a result of these waves, the activity of cells that are close together in space is more strongly correlated than the activity of cells at more distant locations (Feller, Wellis, Stellwagen, Werblin, & Shatz, 1996; McLaughlin et al., 2003). While this particular feature of spontaneous activity is hypothesized to drive visual map formation, the mechanisms by which this is achieved remain elusive (Butts, 2002). One proposed way in which local correlations in the activity of RGCs may contribute to map formation is through a form of neural plasticity that can take advantage of the spatial information provided by spontaneous waves. Such plasticity has received both experimental and theoretical support (Butts, 2002), and is proposed to involve the following steps: (1) retinal neurons located near one another fire bursts of action potentials in close temporal contiguity; (2) this temporal information is relayed to the SC and converted into spatial information through a Hebbian principle (essentially, retinal cells that fire together wire together in the SC). If a Hebbian principle is indeed involved in early map formation,

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Fig. 1. The early development of retinocollicular projections relies on both molecular markers and patterned electrical activity. (a) Prior to the establishment of synaptic connections between retinal axons and their targets in the superior colliculus, a crude topography is established by genetically-expressed markers (e.g., Eph/ephrin molecular guidance cues, (Pfeiffenberger et al., 2006). When gap-junction couplings are established, highly correlated patterns of retinal activity emerge and propagate to the superior colliculus. This period of prenatal development (termed ‘‘pre-critical period’’) is followed by an experience-dependent refinement of the visual system (‘‘critical period’’). (Adapted from: (Khazipov & Luhmann, 2006). (b) In normal (wild-type) animals, spontaneous waves of locally-correlated activity propagate across the retina; in β 2-/- mice, spontaneous activity persists, but is not correlated. (Adapted from: Chandrasekaran et al. (2005)).

a position that is well-supported experimentally (Cline, 2003), then retinal waves constitute the ideal form of activity to drive the development of neural circuits (Butts & Rokhsar, 2001; Firth, Wang, & Feller, 2005). The most convincing experimental evidence linking the statistics of spontaneous retinal wave to the development of axonal projections comes from studies of β 2 knock-out mice. These mice are deficient for the β 2 subunit of the neuronal nicotinic acetylcholine receptor (nAChR), and consequently show an impaired transmission of activation from cholinergic amacrine cells to RGCs. By virtue of projecting to several neighboring RGCs, cholinergic amacrine cells are normally responsible for generating correlated waves of retinal activity; β 2 knock-outs, however, lack these spatial correlations (see Fig. 1b) (Feller, 2002; Xu et al., 1999). As a consequence, the retinocollicular projections of β 2 knock-out mice fail to refine their anatomical position, and remain widely dispersed in the SC (Chandrasekaran, Plaas, Gonzalez, & Crair, 2005; McLaughlin et al., 2003; Mrsic-Flogel et al., 2005). This suggests that the refinement of axonal projections is not driven merely by the amount of activity present, but that specific non-random statistics of spontaneous activity are required to provide adequate map formation (Stellwagen & Shatz, 2002). However, despite strong experimental (McLaughlin et al., 2003) and theoretical evidence (Butts & Rokhsar, 2001) to suggest that spontaneous waves carry important information for the development of neural circuits, particular experimental setbacks prevent a definitive answer to whether patterned activity itself is essential for adequate map formation (Torborg & Feller, 2005). For instance, animals that have been genetically altered to block correlated activity (i.e., β 2 subunit knockout mice) (Chandrasekaran et al., 2005; McLaughlin et al., 2003; Pfeiffenberger, Yamada, & Feldheim, 2006) have decreased firing rates of retinal neurons relative to wild-type mice (5 Hz vs. 20 Hz). In addition, the percentage of retinal cells that are spontaneously active is significantly lower in β 2 knockouts than in their wildtype counterparts. These differences constitute alternatives to an explanation of mistargetted projections based on precise patterned activity. In the present study, computer modeling is employed to investigate the role of patterned retinal activity in map formation. More precisely, the following question is examined: during development of the visual system, are specific aspects of spontaneous retinal waves required for driving the formation of appropriate axonal projections to upstream brain centers (specifically, the SC)? The primary aim is to demonstrate that particular forms of intrin-

sic neural fluctuations are required for the adequate development of neural circuits. A second aim of this study is to employ computer simulations to shed light on the fundamental mechanisms by which synaptic plasticity operates in development. Specifically, the question of whether temporally precise spike-based rules are required for the development of appropriate neural circuitry is a heavily debated issue (Butts, 2002; Butts, Kanold & Shatz, 2007). One the one hand, some argue that immature developing neurons do not possess the fine temporal resolution required for spiketiming-dependent plasticity (STDP) (Ramoa & Mccormick, 1994). In addition, developing synapses exhibit high levels of noise, and are therefore unreliable at relaying information on individual spike potentials (Lisman, 1997). On the other hand, recent theoretical and experimental work suggests that spike-based rules are better suited than less temporally precise (i.e., correlation-based) rules for map plasticity (Young et al., 2007); furthermore, a combined theoretical and experimental examination of early plasticity argues that STDP can account for several findings relative to activity-based refinement of the early visual system (Butts et al., 2007). Here, spike-based and correlation-based forms of plasticity are compared to determine which is more apt at performing map refinement of retinal projections. The remaining sections of this article describe a computational account of activity-driven map formation that combines a wellestablished model of spontaneous retinal waves (Butts, Feller, Shatz, & Rokhsar, 1999; Feller, Butts, Aaron, Rokhsar, & Shatz, 1997) with either STDP or a correlation-based rule for plasticity. Simulation results are discussed in terms of the ability of different forms of waves and different rules for plasticity to perform refinement of a retinocollicular map. 2. Methods In order to examine whether particular statistics of spontaneous activity play a role in retinotopic map formation, we employ a computational model that combines spontaneous waves and synaptic plasticity. This model has two components: (1) spontaneous waves generated according to a biologically realistic model (Feller et al., 1997); and (2) synaptic plasticity (either STDP or a correlation-based rule) modulating the strength of connections between retinal and collicular cells. Each of these two components is discussed in turn. 2.1. Capturing the dynamics of spontaneous waves The model employed here to capture the statistics of spontaneous waves is a well-established account that mimics several as-

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Fig. 2. A model of spontaneous retinal waves. The model contains a layer of amacrine cells, a layer of retinal ganglion cells (RGCs), and a layer of cells from the superior colliculus. A subset of connections is shown, and illustrates between-layer connectivity from amacrine to RGCs, as well as within-layer connectivity in amacrine cells. Only feedforward connections exist between RGCs and SC cells; these projections have a strong initial bias towards topography and can be adjusted through synaptic plasticity (all other connections are fixed). Activation of a certain number of amacrine cells (shown in red) propagates to the retinal ganglion layer, and creates a wave of activity among neighboring cells (shown in blue); this activity propagates forward to cells of the SC. Spontaneously active amacrine cells that do not contribute to the wave are shown in gray. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

pects of retinal activity, including: the low temporal overlap of independent waves; the periodicity at which waves arise; the velocity of wave propagation; and the size of spatial domains covered by waves (Butts et al., 1999; Feller et al., 1997). In this model, waves are generated through the interaction of two layers of neurons, representing amacrine and RGCs (Elliott & Shadbolt, 1999; Feller et al., 1997); (Fig. 2). Amacrine cells receive their input from each other, and send their output to RGCs; the latter do not receive input from each other, and do not send activity back to the amacrine cells. When a wave of activity is generated, it is relayed from the retinal layer to cells of the SC. In accordance with known neuroanatomy, the projections from RGCs to SC cells are strictly feedforward. The strength of RGC–SC projections can be modified through synaptic plasticity (all other connections are fixed, as discussed below). At each time-step t, the membrane potentials of amacrine and retinal RGCs cells have the following wave dynamics: XiRGC (t ) = XiRGC (t − 1) · exp (−1t /τRGC ) + NAC , XiAC

(t ) =

XiAC

(1)

(t − 1) · exp (−1t /τAC ) + NAC ,

where i indexes different cells of the RGC and amacrine populations; 1t is the time-step of the model; NAC corresponds to the number of amacrine cells that fired at the previous time-step (t − 1) within the spatial region φAC ; τRGC and τAC are the integration times of RGCs and amacrine cells, respectively (see Table 1 for a full list of parameters and associated values). In Eq. (1), the amacrine cell layer communicates with the RGC layer only via the total number of amacrine cells active (NAC ); neither amacrine–RGC nor amacrine–amacrine weights are modifiable, and no explicit weight value for these connections is required in the model. Both RGC and amacrine layers are represented as two-dimensional, with 96 × 70 cells and 48 × 35 cells, respectively. Within the amacrine and retinal ganglion layers, a cell fires if any one of the two following conditions is met: 1. its membrane potential is above threshold (θAC ), 2. the cell is within an UP state (lasting for TUP ). XiAC (t ) and XiRGC (t ) are initialized randomly in the range [0, 100] mV. Initially, all cells are in a DOWN state — that is, they are inactive. Within this DOWN state, the potential of cells is clamped AC at zero, so that cells cannot fire for a duration of at least TDOWN (for

RGC amacrine cells) or TDOWN (for RGCs). Once this duration has passed, cells can switch to an UP state if they are either stimulated by other cells to which they are connected, or if they fire spontaneously (with a fixed probability p). Cells can only remain in the UP state for a maximum time duration of TUP , after which they fall back to a DOWN state.

2.2. Synaptic plasticity There is little ambiguity that STDP emerges as the primary rule for long-term plasticity in the early visual system (Fox & Wong, 2005; Mu & Poo, 2006; Zhang, Tao, Holt, Harris, & Poo, 1998). In essence, STDP induces long-term potentiation (LTP) when a presynaptic spike closely precedes an excitatory postsynaptic potential (EPSP), and induces long-term depression (LTD) in the reverse scenario, when an EPSP precedes a presynaptic spike. Thus, the general goal of STDP is to strengthen the presynaptic potentials that best predict the EPSPs. In addition, STDP provides a natural and implicit form of synaptic competition, a mechanism that is essential to the development of visual projections (Sengpiel & Kind, 2002; Tao, Zhang, Bi, & Poo, 2000). In the model examined here, the spike timings of individual RGCs and SC cells are computed by a leaky integrate-and-fire model, as in earlier work (Thivierge, Rivest, & Monchi, 2007), with the following dynamics of membrane potential for RGCs: ViRGC (t ) = α XiRGC (t ) + Vrest + λRGC i

(2)

and for SC cells: VjSC (t ) = Vrest + λSC j +

N X

wij KiSC

(3)

i =1

where retinal ganglion cells (G) and SC cells (S) are indexed i and j respectively, Vrest is the membrane resting potential, wij is a connection efficacy from cell i (retinal ganglion cell) to cell j (SC) and λ is normally-distributed stochastic noise in the range [0, 1]. K is the excitatory potential of incoming spikes from the SC (Gütig & Sompolinsky, 2006; Thivierge & Cisek, 2008): KiSC

= V0 ·

F X j =1

exp

tjSC − t

τfall

! − exp

tjSC − t

τrise

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! .

(4)

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Table 1 Default parameters of the model. Waves of the retinal ganglion cells

Description

1t = 0.1 s TUP = 1.0 s τAC and τRGC = 0.1 s RAC = 120 µm

Time-step Duration of UP state Integration time of amacrine and RGCs Amacrine cell input radius Duration of RGC and amacrine DOWN state

RGC AC TDOWN and TDOWN : Initialized randomly from a Gaussian with µ = 120 s and σ = 0.4 λAC = 34 µm λRGC = 17 µm p = 0.03 θAC = 3.5 θRGC = 3.5 φAC = RAC /λAC

Distance between amacrine cells Distance between RGCs Probability of spontaneous firing Amacrine cell firing threshold RGC firing threshold Number of cells sending inputs to every given amacrine cell

Retinocollicular plasticity α=100

Scaling factor for spontaneous waves Time-window for LTP Time-window for LTD Magnitude of LTP change Magnitude of LTD change Magnitude of potential contributed by each incoming spike Spike amplitude Resting potential Absolute refractory period Spike duration Magnitude of change in synaptic efficacy Rise and fall of incoming spike potential

RGC SC Tthresh and Tthresh : for each RGC and SC cell, randomly chosen from a Gaussian with µ = 18.0 ms and σ = 0.33 XiRGC (t ) and XiAC (t ): initialized randomly between [0, 100] mV ViRGC (t ) and VjSC (t ): initialized randomly between [0, 100] mV

Firing threshold for retinal and SC cells respectively

τ+ = 30 ms τ− = 60 W+ = 10 W− = 100 V0 = 4.0 Vspike = 100 µV Vrest = 0.5 µV Trefract = 3 ms Tspike = 1 ms η = 0.5 τrise and τfall : for each SC cell, randomly chosen from a Gaussian with µ = 3.0 ms and σ = 0.33

Spontaneous wave potentials of a cell at time t in the amacrine and retinal ganglion layers respectively Membrane potentials of cells in the retinal ganglion layer and superior colliculus

where tjSC denotes the spike times of up to 10 previous pulses (indexed by j = 1, . . . , F ), and V0 is a free parameter. RGCs and SC cells emit a spike whenever their membrane potential RGC SC exceeds their firing threshold (Tthresh or Tthresh , for RGCs and SC cells respectively) from below. If a spike is triggered, the membrane potential is set to Vspike for a time duration of Tspike , after which it is reset to its resting state Vrest , and held there for a time length of Trefract corresponding to the absolute refractory period. Given that information from RGCs is transferred to the SC only via the excitatory potential K of Eq. (4), the SC receives direct information solely on the spike timings of RGCs, and not on any other factor, including for instance the amplitude of spontaneous retinal waves. In the above equations (Eqs. (1)–(4)), the properties of spontaneous waves are assumed to be independent of the downstream refinement of retinocollicular projections; this is a reasonable assumption given the lack of known feedback connections from the SC to RGCs. Given this assumption, the amount of computer memory required for running the full model of Eqs. (1)–(4) can be alleviated by first running Eq. (1) for the full duration of a simulation (i.e., 30 s in all results presented here), and afterwards running equations (2)–(4). When examining different parametric settings for Eqs. (2)–(4), the same waves can be reused, rather than generated every time, eliminating potential confounds associated with generating different random waves for each condition. Of course, one concern is to insure that the waves employed are not biased in any obvious way. This might be the case for instance if, by chance, waves occurred only in one portion of the simulated retina. This does not appear to be the case in the model, where the overall firing rate of RGCs was 0.25 Hz (standard deviation: 0.13), which is reasonably close to experimental observations of about 0.33 Hz (Shah & Crair, 2008). Unless otherwise stated, synaptic efficacies wij are modified either through STDP (Fig. 3a). The STDP rule employed is as follows (Abbott & Nelson, 2000):

1wij (t ) =



W+ · exp −1tij /τ+
−W− · exp 1tij /τ−




if 1tij > 0 if 1tij ≤ 0

(5)

where 1tij = ti − tj reflects the difference between the last spike arrival times of presynaptic (ti ) and postsynaptic (tj ) cells; W+ and W− control the magnitude of change in synaptic efficacy; τ+ , and τ− control the time-course of plasticity. Weight updates are applied as follows:

wij (t ) = wij (t − 1) + η1wij (t ) , where η is a free parameter. On each update, weights are constrained to the range [0, 100]. To approximate the initial activity-independent map laid out by molecular markers, synaptic efficacies are initialized according to a Gaussian function (centered at topographic locations, with a mean of 1000 and standard deviation of 30), as in previous modeling work (Song & Abbott, 2001). Of course, initializing weights in this way constitutes a highly abstract method of representing early map formation; it does not capture individual axonal branches or dendrites, but rather an overall density of projections across the neural map. Another limitation is that this approach does not consider the ongoing role of genes after the onset of spontaneous waves (Torborg & Feller, 2005). Activity-regulated transcription is known to occur in the central nervous system; for instance, the expression of certain molecular guidance cues is dependent on the periodicity of spontaneous bursts of action potentials (Fields, Lee, & Cohen, 2005). The implication is that our model cannot claim to represent a complete description of retinocollicular map formation. Much remains to be known about the development of the visual system before such a model can be proposed. As an alternative, the current work focuses on one particular factor of central importance, namely the contribution of spontaneous waves to plasticity-driven map formation. The four STDP parameters (W+ , W− , τ+ , and τ− ) are set by an exhaustive search where different versions of the model are run (each for 30 s) on all combinations of values in the range [1, 2, . . . , 100], chosen independently for each parameter. The values that are retained are the ones that yield the highest ratio of topographic to non-topographic density, as described next.

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Fig. 3. Synaptic plasticity based on either STDP (a) or a correlation-based rule (b). (a) STDP rule, showing change in synaptic efficacy (1w) as a function of the timing of preand post-synaptic spikes (Eq. (5)). (b) Correlation-based rule, showing changes in synaptic efficacy (z-axis) as a function of both pre- and post-synaptic membrane potentials (Eq. (9)) (Young et al., 2007). See Table 1 for parameters.

2.3. Ratio of topographic to non-topographic density

2.5. Different conditions of spontaneous waves

In order to determine the degree to which different versions of the proposed model can generate topographically organized maps, we computed a single score describing the ratio of topographic to non-topographic projections. This ratio is an important criterion in the current work, and will be used throughout the Results section to evaluate the performance of different versions of the model in refining the retinocollicular map. It is computed as follows. First, synaptic strengths wij are converted to a distribution between [0, 1]. Then, a certain topographic window φ is considered (e.g., if φ = 1, then the analysis considers only the weights wii , and if φ > 1, then surrounding weights are also considered). Given this window, the density of topographic weights is computed as follows:

In different simulations of the model, two separate parameters, namely p (the probability of spontaneous activity) and θG (the firing threshold of retinal cells) were varied in order to determine the influence of different properties of spontaneous waves on retinotopic map refinement. In previous work, these parameters have been shown to alter the frequency and amplitude of waves (Butts et al., 1999). The different conditions obtained by varying these parameters were as follows (see Results section for ways of generating different forms of waves in the model): (1) normal waves (i.e., approximating the properties of retinal waves); (Butts et al., 1999); (2) large waves (i.e., where each wave implicates a large number of RGCs); (3) small waves (i.e., where each wave implicates a small number of RGCs); (4) hyperactive waves (i.e., large and highly frequent waves); (5) randomized waves (i.e., normal waves that are scrambled across both time and retinal surface); and (6) no waves (i.e., XiRGC fixed at zero). Randomized waves were obtained by shuffling the matrix XiRGC (activity of all RGCs over time, see Eq. (1)) such that the activity of each cell i is randomly reassigned to a different time-step t. This has for effect of ‘‘breaking up’’ the correlated waves of activity among groups of cells. These ‘‘randomized waves’’ were generated prior to running Eqs. (2)–(4) of the model (responsible for activitydriven map refinement). While other means of generating random waves are possible (e.g., by disconnecting AC–RGC connections), our method is the most direct way to disrupt local correlations on the retina while ensuring that other aspects of spontaneous activity, including firing rates, are preserved. In order to determine whether the density of topographic projections (Eq. (8)) was similar or different across conditions with different forms of spontaneous activity, Student’s t-tests were performed where the individual data points corresponded to the density of topographic projections (i.e., diagonal entries wii ). Results were considered statistically reliable if p < 0.01. A similar analysis was performed for non-topographic projections. Finally, analyses comparing the density ratio (Eq. (8)) across different versions of the model were also performed. All simulations were done with MATLAB software. Equations for RGC–amacrine spontaneous waves (Eq. (1)) as well as retinocollicular plasticity (Eqs. (2)–(5) and Eq. (9)) were simulated with an integration time of 1 ms. Unless otherwise indicated, all parameters employed are as listed in Table 1. In keeping with the main goal of the current paper, these parameters were chosen to demonstrate the computational advantage of non-random waves, and are not aimed at fitting precise experimental values.

dtopographic =

(N ,i+φ) N j=min X X

w ¯ j,i /φ

(6)

i=1 j=max(1,i−φ)

for the final distribution of weights w ¯ j,i (where the over-hat indicates normalization). The density of non-topographic weights is obtained as follows: dnon-topographic =

N X N X


w ¯ j,i − dtopographic / (N − φ)

(7)

i=1 j=1

where N is the total number of retinal cells. Finally, the following ratio is calculated in order to determine the relative densities of topographic and non-topographic projections: FINAL dratio = dFINAL topographic /dnon-topographic .

(8)

The value of dratio is highest when topographic projections have a high density and non-topographic projections have a low density. The above Eqs. (6)–(8) are computed a total of 50 times for different widths spanning between 1 and 50 cells around the exact topographic locations (wii ). Results reported are based on optimal parameters (i.e., yielding a maximal value of dratio after averaging over all retinal windows). 2.4. Correlation-based learning As an alternative to STDP (Eq. (5)), we also considered a correlation-based rule (Thivierge et al., 2007; Young et al., 2007):

1wij (t ) = η · ViRGC (t ) · VjSC (t ) .

(9)

Unlike STDP, the above rule is not based on precise spike timings of RGCs and SC cells; rather, it is based on ongoing fluctuations in pre- and post-synaptic activity. As another crucial distinction, the correlation-based rule does not exhibit the asymmetry in LTP/LTD that is a defining feature of STDP (see Fig. 3b).

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Fig. 4. Wave-like activity of retinal ganglion cells evolving through time. A total of 1.5 contiguous seconds of simulation are displayed (moving from left to right, and top to bottom, beginning in the upmost left corner). Color maps depict the activation of retinal cells in the middle of each time frame of activity (each frame spans 100 ms). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Results

3.2. Refinement of retinocollicular projections

3.1. The dynamics of spontaneous waves

Do these different forms of spontaneous activity lead to different outcomes in terms of map refinement when combined with synaptic plasticity? After 30 s of simulation combining spontaneous waves and synaptic plasticity, different outcomes emerged when the properties of waves were manipulated. Here, we define a successful refinement as a process that maintains topographic projections, but significantly reduces nontopographic projections (see Eq. (8)). Overall, results argue that different forms of spontaneous activity lead to statistically reliable differences in terms of map formation. This finding is particularly well illustrated in the following simulations. If normal waves are randomly scrambled in time and across cells (i.e., ‘‘randomized waves’’ condition, see Methods section for details), both topographic (t (93) = 3.75, p < 0.001) and non-topographic (t (93) = 15.22, p < 0.001) connections reduce their strength when compared to the normal condition (Fig. 6a vs. b; see Fig. 6c for mean results). This suggests that retinocollicular formation with random waves lacks specificity and reduces the connection density of different map regions regardless of whether they are adequately linked (i.e., topographic) or not (i.e., non-topographic). To compare the proportion of topographic and non-topographic weights in the correlated and randomized wave conditions, Eq. (8) is computed for different retinal windows (between [1, 50]). Across these different retinal windows, the resulting ratio of topographic to non-topographic weights is overall higher with correlated activity than with random waves (t (49) = 7.79, p < 0.01), resulting in a positive value when the ratio of the

Through interactions between RGCs and amacrine cells, the model produces waves of spontaneous activity that originate from random locations on the retina and sweep across its surface (see Fig. 4). In other work, several properties of these simulated waves have been shown to capture the key aspects of retinal activity observed experimentally, including large wavefronts, slow propagating speed across the retinal surface, long inter-wave time interval, and random location of origin of different waves (Butts et al., 1999; Feller et al., 1997). By adjusting two parameters in the model, namely p (the probability of spontaneous firings) and θG (the threshold of activity required for firing), it is possible to qualitatively vary the dynamics of spontaneous waves (see Fig. 5 for a ‘‘phase diagram’’ showing different wave properties over the p–θG parameter space). With a small probability of random activity (p), waves become more active (Fig. 5b, ‘‘large waves’’). By increasing spontaneous firings (p), waves become less frequent (Fig. 5b, ‘‘small waves’’), because there is a higher probability of individual cells firing and therefore a lower probability of these cells forming waves of adequate size. Finally, by decreasing the firing threshold (θG ), waves become both more frequent and more active (Fig. 5b, ‘‘hyperactive waves’’). These different manipulations, altering the properties of spontaneous waves, were directly used to examine the effect of spontaneous activity on map refinement, as discussed next.

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Fig. 5. The dynamics of spontaneous waves can be altered by adjusting two parameters of the model. (a) Changes in the probability of spontaneous activity (p) and firing threshold of retinal cells (θG ) lead to qualitative differences in the waves produced by the model (from: Butts et al. (1999). (b) Percentage of cells active as a function of time for different types of spontaneous waves. A cell is considered active if its membrane potential crosses the firing threshold (see Methods for details).

latter condition is subtracted from that of the former (Fig. 6d, in a range of retinal windows between [1, 40]). These results are particularly convincing in demonstrating that spatial correlations among active retinal cells can directly influence the refinement of retinocollicular projections, even when other factors such as overall firing rate are kept constant. Similar results are found when wave statistics are modified in order to generate smaller, larger, or hyperactive waves (Fig. 7a, for mean weights see Fig. 7b). If smaller waves are generated (condition III in Fig. 5b), significantly less non-topographic projections are maintained when compared to a condition with normal waves (t (93) = 15.226, p < 0.001), while a comparable density of topographic projections is maintained (t (93) = 2.31, p > 0.02). If larger waves are generated (condition IV in Fig. 5b), both topographic (t (93) = 3.13, p < 0.002) and non-topographic (t (93) = 3.74, p < 0.001) projections reduce their density when compared to the normal condition. A similar result is found when hyperactive waves are generated (condition II in Fig. 5b), such that both topographic (t (93) = 3.59, p < 0.001) and non-topographic (t (93) = 40.59, p < 0.001) connections reduce their strength. Finally, in a condition that completely eliminates spontaneous activity, the densities of topographic and non-topographic projections do not depart from initial conditions (i.e., there is no change in the map). Taken together, the above simulation results corroborate experimental evidence arguing for the importance of wave properties in driving neural development (McLaughlin et al., 2003). In addition, they suggest that if spontaneous activity could be completely abolished experimentally while other factors are kept intact, the pruning of aberrant projections would be drastically diminished. 3.3. Influence of spike-timing precision on map formation When a presynaptic pulse and an EPSP occur in close temporal contiguity, there is a window of a few milliseconds in either

direction where STDP seems unable to distinguish the separate spike arrival times. A recent quantitative analysis of STDP clearly shows this point (Thivierge et al., 2007): when pre- and postsynaptic signals arrive within about 5 ms of each other, both LTP and LTD can occur, in a seemingly random fashion. Given this limit on the temporal precision of STDP, it is critical to determine whether our model is dependent upon a precise (i.e., 1 ms resolution) recognition of spike arrival times in order to perform map refinement. This is tested by using a modified version of the STDP rule, where spikes that occur within a short temporal interval of each other lead to a random outcome of either LTP or LTD (Fig. 8a). By varying this temporal interval (1 ms, 5 ms, and 10 ms), we can estimate how the precision of the STDP affects map refinement. Results show a capacity of the model to perform map refinement even when the temporal resolution of STDP is reduced from 1 ms to 5 ms (Fig. 8b). Indeed, following 30 s of simulation combining STDP with spontaneous retinal waves, no statistical differences are found between 1 ms and 5 ms resolution in terms of topographic density (t (93) = 0.008, p > 0.99) or nontopographic density (t (93) = 1.81, p > 0.07). However, when further lowering the resolution to 10 ms (i.e., randomly generating LTP or LTD whenever spike arrival times are <10 ms apart), results worsen: the density of both topographic (t (93) = 10.22, p < 0.001) and non-topographic (t (93) = 12.52, p < 0.001) projections are significantly lower with 10 ms resolution than with 1 ms resolution. Hence, lowering the temporal resolution to 10 ms leads to a lack of specificity in the refinement process, as both topographic and non-topographic projections are reduced. Despite the disrupted performances of the STDP rule with 10 ms of resolution, this rule is still able to maintain a higher density of topographic projections compared to non-topographic projections; several other versions of the model (e.g., Figs. 6 and 7) did not. In sum, the proposed model of synaptic plasticity can perform an adequate map refinement (i.e., maintaining topographic

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Fig. 6. Randomly shuffling spontaneous waves in time and in space disrupts map formation. (a)–(b) Maps obtained with normal and randomly shuffled waves following 30 s of simulation. The top figure of each panel shows two-dimensional synaptic weights from retina (x-axis) to superior colliculus (y-axis). These weights are normalized between [0, 1] (see color bar at right of figure). The bottom figure of each panel shows a single synaptic weight (the retinal position from which this weight originates is indicated by an arrow in the top panel). solid line: original distribution of weights (representing neural density) prior to activity-dependent refinement. dashed line: distribution following 30 s of simulation. (c) Means of topographic and non-topographic weights resulting from normal and randomized waves of activity. (d) Difference in topographic/non-topographic ratios (Eq. (8)) between normal and randomized waves, for different windows of retina. These values are obtained by first computing Eqs. (6)–(8) independently for the normal and random wave conditions, then subtracting the ratio of the latter from that of the former. Positive values indicate that the refinement obtained with normal waves surpassed that obtained with random waves. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

projections while reducing non-topographic projections) even when it cannot discriminate spikes arriving within <5 ms of each other. However, lowering the temporal resolution further (i.e., <10 ms) hinders the capacity of the model to adequately refine its topographic map. These results argue that while temporal precision in the order of 1 ms may not be necessary for the development of retinocollicular projections, some degree of precision may nonetheless be required in order to extract useful information from the properties of spontaneous waves. 3.4. Comparing spike-based and correlation-based plasticity An important goal of this study is to determine if certain rules for synaptic plasticity were more efficient than others at providing refinement. When comparing STDP with a correlation-based rule, the latter yielded markedly worse refinement capabilities (Fig. 9a vs. b; see Fig. 9c for mean results): compared to the STDP rule, the correlation-based rule decreased topographic projections (t (93) =

3.14, p < 0.002) and increased non-topographic projections (t (93) = 228.68, p < 0.01). A striking effect of the correlation-based rule is that a large portion of connections are pushed to their maximal value (i.e., 1.0), as indicated by the white areas in Fig. 9a. The fact that the correlation-based rule indiscriminately increases the strength of both topographic and non-topographic projections shows its lack of specificity in the refinement process. By comparison, a rule based on STDP has the ability to decrease non-topographic connections while preserving topographic connections (Fig. 6a). The ratio of topographic versus non-topographic weights (Eq. (8)) was significantly larger for STDP than for correlation-based plasticity (t (49) = 54.77, p < 0.01), resulting in a positive value when the ratio obtained with the latter condition is subtracted from the ratio obtained with the former (Fig. 9d). These results corroborate recent evidence arguing for the importance of spikebased plasticity in map formation and reorganization (Young et al., 2007).

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Fig. 7. The refinement of retinocollicular projections is disrupted when different properties of spontaneous waves are altered. (a) Top panel: Synaptic weights from retina to superior colliculus, normalized between [0, 1] (see color bar at right of figure). Bottom panel: Single synaptic weight (retinal position shown by arrow in top panel) for initial distribution (solid line) and distribution following activity-dependent refinement (dashed line). See Fig. 5a for the different wave parameters employed. (b) Average synaptic weights of topographic and non-topographic projections resulting from 30 s of simulation with either normal waves, smaller waves, hyperactive waves, or larger waves (see Fig. 5 for details of these different waves). Vertical bars: SEM.

4. Discussion This paper examined the following question: do the specific properties of spontaneous waves of activity play a role in shaping axonal projections during the early stages of development? This question was addressed by combining a well-established computational model of retinal spontaneous activity (Butts et al., 1999; Feller et al., 1997) with a rule for synaptic plasticity (based on STDP) documented in the development of the early visual system (Mu & Poo, 2006; Zhang et al., 1998). Our results are clear: early map formation depends strongly upon the particular statistics of spontaneous activity to eliminate extranumerary projections in inappropriate termination zones of the SC. This finding is consistent with both experimental and theoretical evidence: 1. the information content of spontaneous waves varies with respect to the spatial position of retinal cells (Butts & Rokhsar, 2001); 2. disrupting correlated activity through genetic knockout of the β 2 subunit (Chandrasekaran et al., 2005; McLaughlin et al., 2003; Pfeiffenberger et al., 2006) attenuates activitydependent map refinement, and results in the maintenance of a higher density of aberrant (non-topographic) projections when compared to normal animals.

The current study goes beyond the above findings in suggesting that correlated activity may exert a direct influence on the refinement of retinocollicular projections. The idea of a direct link between patterned activity and retinotopic refinement, although highly plausible, is difficult to confirm experimentally because current methods for altering correlated activity also alter other aspects of spontaneous activity, leading to decreased firing rates and a lower proportion of active retinal cells (Torborg & Feller, 2005). It is also difficult at present to determine whether a link between patterned activity and neural development exists in other structures. For instance, suggestions of a potential involvement of patterned activity in the segregation of retinogenicular axons into eye-specific layers are still a matter of debate (Firth et al., 2005; Torborg & Feller, 2005). In addition to highlighting the role of patterned activity, our study also argues that certain rules for synaptic plasticity (e.g., STDP) are better at driving retinotopic map formation than others (e.g., a correlation-based rule). In simulations, a rule based on precise spike interactions (i.e., in the range of a few milliseconds) more effectively reduces the strength of projections to non-topographic regions when compared to a rule based on correlations that extend over a longer time-course. Of course, these results cannot definitively eliminate alternative rules for plasticity; however, recent work suggests that STDP does explain experimental results on developing neural circuits (Butts et al., 2007) as well as reorganization following damage (Young et al., 2007).

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Fig. 8. Map formation under spike-timing-dependent plasticity is robust to lower degrees of resolution in spike detection. (a) Different versions of spike-timing-dependent plasticity where the degree of spike resolution is altered. With 1 ms accuracy, the rule correctly assigns all spike events (represented by black circles) to either LTP (positive change in weight, 1w > 0) or LTD (negative change in weight, 1w < 0), even when these events are merely 1 ms apart. With 5 ms and 10 ms accuracy, spike events that occur too close together in time (shown in red) are randomly assigned to either LTP or LTD. The temporal interval between pre- and post-synaptic spikes is shown on the x-axis (1t, see Eq. (5)). (b) Average synaptic weights of topographic and non-topographic projections resulting from 30 s of simulation with either 1 ms, 5 ms, or 10 ms of spike-timing accuracy. Vertical bars: SEM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The success of STDP in refining axonal projections may be due to a mechanism of ‘‘didactic reorganization’’. (Song & Abbott, 2001; Young et al., 2007). Accordingly, if a retinal ganglion cell ‘A’ causes a neuron ‘B’ of the SC to fire, the connection from A to B will be potentiated. As a result, the behavior of neuron A will become more similar to that of neuron B; in addition, any other RGC contributing to the response of B will also be potentiated, and increase its similarity to B. When correlated activity is present on the retina, neighboring regions of neural tissue will propagate to the SC and activate nearby targets. This activation will be reinforced through potentiation, therefore preserving spatial relationships from the RGC to the SC layer, and resulting in a topographically ordered map. Alternatively, the correlation-based rule differs from STDP in that the connection from A to B can be potentiated regardless of whether retinal ganglion neuron A or SC neuron B fired first. As a result, correlation-based learning potentiates the connection to SC cells that are not directly caused by the activity of a particular retinal ganglion cell, hence disrupting topography. Adding further plausibility to the role of STDP in map formation, our simulation results suggest that even when the model cannot extract spike information with <5 ms of precision, it can still perform an adequate refinement of axonal projections. This finding is crucial to the biological relevance of the model, as STDP reportedly fails to discriminate spike events less than about 5 ms apart (Thivierge et al., 2007). Furthermore, the ability of the model to perform refinement despite a rigorous temporal precision agrees well with the lack of fine precision found in immature developing neurons. Because of factors including high levels of noise (Lisman, 1997) and slow NMDA currents (Ramoa & Mccormick, 1994), developing cells are too sluggish to enable a precise spike timing code; hence, a plausible rule for synaptic

plasticity must be robust to some degree of imprecision regarding spike information, as captured by the proposed model. Because our simulations were limited to 30 s of spontaneous activity, one suggestion might be that different wave statistics merely slow down map refinement without preventing it entirely. While this possibility cannot be entirely dismissed, an important consideration is that spontaneous waves influence map formation during a time-delimited period of development on the order of only a few days (Feldman, Nicoll, Malenka, & Isaac, 1998; Firth et al., 2005; McLaughlin et al., 2003). This short time window may elicit a developmental pressure that favors rapid mechanisms of map refinement over slower alternatives. While the current study addressed the role of patterned activity in neural development, it is clear that highly non-random forms of spontaneous activity also contribute to cognitive and behavioral processes in adulthood (Fox et al., 2007; Pessoa et al., 2002; Sapir et al., 2005). The coming years will likely witness the emergence of new and more sophisticated models that link spontaneous fluctuations in brain activity to cognitive operations (Freeman, 2007), as well as models that bridge the genetically-driven development of neural circuits and the capacity for producing complex mental operations (Thivierge & Marcus, 2007). Acknowledgments This work was supported by a Fellowship from the Fonds Québéçois de Recherche sur la Nature et les Technologies (FQRNT). The author is grateful to Evan Balaban (McGill University) as well as two anonymous reviewers for comments the manuscript.

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Fig. 9. Comparing a spike-based rule with a correlation-based rule for activity-dependent refinement. (a) Synaptic weights from retinal to superior colliculus after 30 s of simulation using correlation-based plasticity. (b) Single synaptic weight for initial position (solid line) and following refinement (dashed line). (c) Means of topographic and non-topographic weights resulting from spike-based and correlation-based plasticity. (d) Difference in topographic/non-topographic ratios (Eq. (8)) between spike-based and correlation-based rules, for different windows of retina (done in the same way as Fig. 6d).

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