Neural dynamics within recurrent cortical networks is an important component of neural processing. However, the learning rules that allow networks composed of hundreds or thousands of recurrently connected neurons to develop stable dynamical states are poorly understood. Here I use a neural network model to examine the emergence of stable dynamical states within recurrent networks. I describe a learning rule that can account both for the development of stable dynamics and guide networks to states that have been observed experimentally, specifically, states that instantiate a sparse code for time. Across trials, each neuron fires during a specific time window; by connecting the neurons to a hypothetical set of output units, it is possible to generate arbitrary spatial-temporal output patterns. Intertrial jitter of the spike time of a given neuron increases as a direct function of the delay at which it fires. These results establish a learning rule by which cortical networks can potentially process temporal information in a self-organizing manner, in the absence of specialized timing mechanisms.
Local dynamics within cortical networks play a fundamental role in neural computations (Koch and Fuster 1989; Ringach et al. 1997; Somers et al. 1995). Dynamics in the form of complex spatial-temporal patterns of neuronal firing have been observed in vitro (Beggs and Plenz 2004; Buonomano 2003; Ikegaya et al. 2004; Jimbo et al. 1999) and in vivo and been shown to contain information about sensory stimuli (Gawne et al. 1996; Laurent et al. 1996) and motor behavior (Hahnloser et al. 2002; Weesberg et al. 2000). While it has been established that networks exhibit complex spatial-temporal dynamics, the synaptic learning rules that allow recurrent networks to develop functional and stable dynamics are not known. Indeed, conventional coincident-based learning rules are unstable (Miller and MacKay 1994; Turrigiano and Nelson 2004) and can lead to runaway excitation in recurrent networks.
A potential computational function of spatial-temporal patterns of activity is temporal processing and motor control. Decoding temporal information, or generating timed responses on the order of tens to hundreds of milliseconds, is a fundamental component of many sensory and motor tasks (Mauk and Buonomano 2004). Indeed the precise sequential generation of motor responses is a virtually ubiquitous component of behavior. One of the most studied forms of complex sensory-motor processing is the birdsong system (Bottjer and Arnold 1997; Doupe and Kuhl 1999). Song generation relies on precisely timed sequential generation of motor patterns over both the time scale of individual syllable features and sequences of syllables (Fee et al. 2004). While relatively little is known about the neural mechanisms underling the generation of precisely timed motor sequences, it has recently been shown that there is a sparse code for time in the premotor area HVc, which may control song production (Hahnloser et al. 2002). Dynamically changing patterns of activity have also been proposed to code for time in the cerebellum and underlie certain motor patterns (Medina et al. 2000). Sparse long-lasting responses have also been observed in vitro. As shown in Fig. 1 in cortical organotypic cultures, a single stimulus can elicit single spikes at latencies of a few hundred milliseconds (Buonomano 2003). These slices contain thousands of recurrently connected neurons and initially exhibit weak synaptic connections (Echevarría and Albus 2000; Muller et al. 1993). In this study, I examined how dynamics may emerge in this general class of networks.
It has been shown that a modified form of synaptic scaling can guide networks to a stable dynamical state. Interestingly, the dynamical states that emerge from this learning rule allow the network to generate a sparse temporal code for time. Qualitatively the architecture of the model and dynamical states observed are consistent with experimental data from cortical organotypic slices (Fig. 1) (Buonomano 2003).
All simulations were performed with NEURON (Hines and Carnevale 1997). Each neuron was simulated as a single compartment integrate-and-fire unit, with dynamic synapses. The ratio of excitatory (Ex) to inhibitory (Inh) neurons and connectivity probability was based on experimental data (Beaulieu et al. 1992). Specifically, 80% of the units were excitatory and 20% inhibitory. In our default simulations, there were 320 Ex and 80 Inh units (each Ex unit received 20 excitatory and 5 inhibitory synapses, and each Inh unit received 5 excitatory synapses). Connectivity was uniformly random.
The resting membrane potential of all units was –60 mV. Thresholds were set from a normal distribution (σ 2 = 5% of mean threshold); the mean thresholds for the Ex and Inh units were –40 and –45 mV, respectively. After a spike, the voltage was reset to –60 and –65 for the Ex and Inh units, respectively. Membrane time constants were 30 ms for the Ex units and 10 ms for the Inh units. Input resistance was 300 MΩ. See supplementary information1 for further details.
α-Amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA), N-methyl-d-aspartate (NMDA), and GABAA synaptic currents were simulated using a kinetic model (Buonomano 2000; Destexhe et al. 1994; Karmarkar and Buonomano 2002) (supplemental information). Short-term plasticity was incorporated in all the synapses based on experimental data and implemented according to Markram et al. (1998). Specifically the Ex → Ex synapses exhibited depression—U (use of synaptic efficacy parameter) = 0.5; τrec = 700 ms—Ex → Inh synapses exhibited facilitation: U = 0.2; τrec = 125 ms; τfac = 500 ms. Inh → Ex dynamics were based on basket cell synapses: U = 0.25; τrec = 700 ms; τfac = 25 ms (Gupta et al. 2000). The presence of short-term plasticity was not essential to the behavior of the networks described below. Initial synaptic strengths were chosen from a normal distribution. For the Ex → Ex, Ex → Inh, and Inh → Ex synapses, the mean initial synaptic strength was 0.5, 2, and 8 μS, respectively. To insure that plasticity did not result in an unphysiological state in which a single presynaptic neuron could fire a postsynaptic neuron, the maximal synaptic weight for the Ex → Ex and Ex → Inh connections was set at 10 and 30 μS, respectively. The NMDA current contributed to long latency responses but was not essential for the stability or dynamics of the network. The strength of the NMDA component was a fixed ratio of the AMPA strength (0.3). The strength of the inhibitory synapses was fixed throughout the simulations. Results were robust to large changes in initial parameters. Changes in an order of magnitude in the Inh → Ex did not alter the final homeostatic state. The learning rate α was set to 0.05 in the simulations presented here.
The variable Ai measures the average activity of neuron i and is defined as (1) where Si(t) represents the presence (1) or absence (0) of a spike at time t. I arbitrarily used τ = 10 s and a stimulus presentation rate (intertrial interval) of 10 s (see discussion)
The synaptic scaling learning rule is generally represented as (van Rossum et al. 2000) (2) Wpre,post represents the synaptic strength between a presynaptic and postsynaptic neuron and is updated at the end of each trial. Unless stated otherwise, Agoal was set at 1 and 2 for the Ex and Inh units, respectively. Agoal was set higher for the Inh units, because inhibitory neurons have higher evoked and spontaneous firing rates (Swadlow 1989).
Information measure of timing accuracy
We quantified the information present in the spatial pattern of spikes of the Ex units by determining the information content of the current time bin (Borst and Theunissen 1999). In essence, we can think of each time bin as a stimulus, and ask what is the mutual information between the spike patterns (in time bin n) and time bin n. However, it was of interest to know not only the mutual information of the system over all time bins but in relation to each time bin, to determine if there is more information available at certain intervals. Thus I asked what is the information about time bin n in relation to all time bins ≠ n. The mutual information about time bin n was equal to (3) where H(S) is the conventional entropy measure of all spike patterns, H(n) reflects the entropy of being in time been n versus all other time bins, and thus is the same for all time bins and based on p(1/N) and p(1 − 1/N), where N is the total number of time bins. H(S,n) is the joint entropy. Because the occurrence of time bin n and all other time bins are not equiprobable, the maximal mutual information at all time steps is equal to (4)
Realistic measures of timing accuracy
To obtain an estimate of the timing ability of the network as a whole that would be available to an output neuron that was trained to fire at a specific time, I incorporated five output units to the network. Each output unit received synapses from all the Ex units. On training trials, the weight between Exi and outputj (where j is the target interval) was increased if i was active at time j. Testing was performed over a separate set of trials.
To address the emergence of stable dynamical states in recurrent networks, I first examined a previously described learning rule–synaptic scaling (Eq. 2; Turrigiano et al. 1998), which has been shown to guide neurons in feed-forward networks to a stable homeostatic state (Turrigiano and Nelson 2004; van Rossum et al. 2000). This experimentally derived learning rule was proposed in the context of maintaining approximately constant levels of activity over the course of days. Here I will examine if synaptic scaling can generalize to a different condition, specifically a condition in which activity is stimulus-driven over a shorter time course and the desired activity patterns are sparse (Hahnloser et al. 2002).
Unless otherwise stated, the artificial neural network was composed of 400 recurrently connected integrate-and-fire neurons, and synapses were initially weak. These general assumptions are meant to loosely reflect those observed in dissociated or organotypic cultures (Muller et al. 1993). The target level of activity was initially set to one spike per trial for the Ex neurons. This assumption was meant to capture the sparse activity often observed in response to a single stimulus in organotypic cultures (Fig. 1) (Buonomano 2003), as well as data from primary sensory areas indicating that neurons generally respond to a single transient stimuli with one spike (e.g., Armstrong-James et al. 1994; Kilgard and Merzenich 1998). Each trial consisted of activating a subset of neurons (input neurons), which again is comparable with extracellular stimulation in an in vitro slice, where a random subset of neurons close to the electrode will function as an input source. The network evolved under the guidance of synaptic scaling over 1,000 trials. Synaptic scaling essentially states that the conductances of all synapses onto a postsynaptic neuron are increased (multiplicatively) if the average activity level (Ai) of the cell is below a predetermined setpoint and decreased if the cell is hyperactive. Figure 2 shows the results from simulations in which the setpoint of the Ex neurons was set to 1 spike/trial (see Movie 1). In the initial trials, activation of the input neurons did not result in any suprathreshold activity in the other neurons. With training, the learning rule was effective in generating network activity. However, it did not converge to a steady state in which neurons stabilized at their target activity level. Instead, oscillatory behavior was observed. This behavior was observed in dozens of stimulations with different initial conditions and independent of the learning rate (see also Fig. 4). Because synaptic scaling is known to be stable in feed-forward networks (Turrigiano and Nelson 2004), I hypothesized the instability arises as a result of the recurrent architecture. As shown in Fig. 3A, using a simplified implementation of the above network, I examined the effects of recurrency. If all postsynaptic Ex neurons received only a single synapse (thus effectively implementing a feed-forward network), each neuron reached it's target level of activity and the network converged. If a minimal degree of recurrency was introduced, by assigning two or four synapses to each neuron, convergence was not observed.
An inherent limitation of synaptic scaling when incorporated into recurrent networks can be schematized in a toy model (Fig. 3B). Consider two reciprocally connected neurons (A and B), one of which (A) receives a synapse from an input neuron. If the synapses onto A from the input (w1) and B (w2) start in an initial condition such that w1 < w2, then because w1 and w2 will always be scaled by the same factor, w1 will always be smaller than w2. However, if the network is to converge to a stable state in which each neuron is active once per trial, clearly w1 must be larger than w2. Although this is simply a toy model, one can see a potential limitation with synaptic scaling. The ratio of all synapses onto a cell will always remain the same because Δw is a function only of postsynaptic activity. In essence, although a cell may have 1,000 synapses, it only has one degree of freedom. The network cannot converge because there is no solution that maintains the synaptic ratios onto a postsynaptic neuron. Thus the lack of convergence is independent of the parameters of the model. While the learning rates and time window over which activity is calculated can alter the magnitude and period of the oscillations, the synaptic ratios remain unchanged; thus a solution cannot be reached.
To allow a neuron to change the relative synaptic strength ratios during training, the synaptic scaling rule was modified to include a presynaptic component (5) Under this rule, a postsynaptic neuron will preferentially potentiate synapses from neurons that are more active (independent of when it was active within a trial), I will refer to Eq. 4 as presynaptic-dependent scaling.
Figure 4, A and B, shows the results from a simulation using the same initial parameters used in the simulations shown in Fig. 2. With training, both the neurons and global network activity converge to a stable state (Movie 2). Each Ex neuron fired once per trial, and the rate of change of synaptic weights approached zero (data not shown). Figure 2A also shows that there is complex spatial-temporal structure to the network dynamics. By plotting the same data sorted by latency (Fig. 4B; Movie 3), it is possible to observe a synfire-like pattern, in which activity propagates throughout the network. However, propagation is not based on the spatial arrangement of the neurons, because there is no topography in the network.
To compare the ability of this learning rule to drive the neurons in a recurrent network to their target level of activity with synaptic scaling, I performed simulations using different setpoints. To ensure that a true steady state was reached, 6,000 trials were performed, and the mean level of activity and stability of the last 1,000 were quantified. Figure 4C shows that both learning rules were effective in bringing mean activity (averaged over all Ex units over 1,000 trials) to near the setpoint. Figure 4D shows the SD of the mean activity level. The high SD in the synaptic scaling simulations reflects the lack of convergence. In contrast, the low SD observed using Eq. 4 reflects the convergence to a stable dynamical state.
As shown in Fig. 4B, presynaptic-dependent scaling produced a spatial-temporal pattern of activity, in which different neurons fired at specific time windows from stimulus onset. This pattern could potentially function as a population code for time. Figure 5A shows a poststimulus time histogram (PSTH) raster of all the neurons in the network after training over 25 trials. Note that across trials spike jitter is a function of latency. This is expected in a system in which spike latency variance is amplified within a trial and has been recently observed in neural networks in vitro (Buonomano 2003). The ability of the network to code for time was quantified in two ways. First, the information available in the spatial pattern of activity was calculated. Three different temporal binnings were used: 1, 4, and 8 ms. Maximal information at a given time step implies that there are spatial patterns that can be used with 100% certainty to determine the time within the accuracy of the bin used. As shown in Fig. 5B, at 8-ms resolution, the network can with certainty determine the time ≤100 ms. However, at 1-ms resolution, there are gaps. The biggest one is at ∼10 ms, in which no neurons are active due to fast inhibition. Thus the network cannot time with millisecond resolution. To quantify the ability of the network to tell time in a physiologically plausible manner, it was assumed that the Ex units provided input to a set of output units. Each of the output units receives inputs from all Ex units and represents an interval from 25 to 125 ms. The synaptic weights from the Ex to the output units were adjusted by training over 25 trials. Performance was subsequently tested using a distinct set of 25 trials. Figure 5C shows the average response of the output units during a single trial. These results show that a network without any specialized timing mechanisms can process temporal information.
Convergence time and propagation speed
The above simulations were performed with a network composed of 400 units. To determine if performance was independent of network size, I examined the performance of the presynaptic-dependent scaling learning rule with networks composed of 200, 400, 800, and 1,200 units (Fig. 6A). All the networks were scaled; that is, the ratio of Ex to Inh units remained constant, as did the probability of connectivity (thus the total number of synapses per neuron increased). Interestingly, the larger network converged faster. This is because, since each neuron has more synapses, the average synaptic strength for a neuron to fire is decreased. Thus fewer trials are required to allow synapses to grow sufficiently to fire a neuron. However, larger networks also tend to overshoot and thus do not converge in a monotonic fashion (for α of 0.05). Again, this is because the total synaptic input grows at a faster rate because of the increased number of synapses.
We also examined the effect of changing the connection probabilities within the network. These simulations were performed in the large network with 1,200 units. The final state of three sets of simulations with three different connection probabilities are shown in Fig. 6B. As expected, the sparser the connectivity, the longer it takes activity to propagate throughout the network, thus extending the intervals the network could code for. However, there is a trade-off in that the intertrial variability also increases as the total duration increases (data not shown).
To examine the effects of any single neuron on overall dynamics of the network, I performed ablation experiments (Fig. 7) —simulations in which 1% of the neurons lesioned revealed little change in the overall spatial-temporal pattern of activity. Lesioning 5% of the cells did not change the mean level of activity but did dramatically change the dynamics. Whereas most cells exhibited a longer latency, some exhibited no shift or a decrease in latency. The shift toward longer latencies is the result of decreased spatial summation, because most cells receive fewer active inputs, thus taking more time to reach threshold. This increased latency can be amplified as activity propagates throughout the network. As shown in Fig. 7A, some neurons can fire earlier. This can occur as a result of disinhibition: for the same reason the Ex neurons may fire later, the Inh neurons will also, allowing a few Ex neurons to fire earlier.
Coding for longer durations
In the preceding simulations, there was no spatial topography; that is, all neurons had equiprobable connection probabilities. To examine the effects of presynaptic-dependent scaling in a more realistic network, I examined the effect of topography. In these simulations, there were 1,200 units. Each Ex unit received inputs from 80 other Ex units. Connection probability was normally distributed around each neuron placed on a one-dimensional array. Figure 8 shows a raster from a single trial after training. As in the previous simulations, presynaptic-dependent scaling produced a homeostatic state in which each Ex neuron fired once during a trial. As a result of the topography, the total propagation time increased to ∼350 ms. To examine the readout of a network with later responses, I trained five output units to respond to intervals of 50, 100, 150, 200, and 250 ms. Consistent with in vitro (Buonomano 2003) and psychophysical data (Karmarkar and Buonomano 2003), the across-trial jitter increased as a function of latency. These results confirm that, with or without spatial topography, presynaptic-dependent scaling produces a stable network state. Additionally, this simulation shows that the time range the network can encode is extended by using larger networks with topography. The maximal latency is extended in the topographic network because of the presence of spatial propagation: neurons distant from the input signal can only begin to fire once their neighbors have begun to fire.
The above results establish a learning rule that can lead a naïve recurrent network to achieve a stable dynamical state that implements a sparse code for time. Qualitatively, the results account for the sparse long-latency responses observed in cortical organotypic slices (Buonomano 2003).
It is well known that the control of activity in spiking recurrent networks is much more difficult then in feed-forward networks (e.g., Maass and Sontag 1999; Pearlmutter 1995). Because synaptic scaling has been shown to lead to stability in feed-forward networks (van Rossum et al. 2000) or contribute to stabilization in recurrent networks (Renart et al. 2003), our first approach was to determine if synaptic scaling could generate stable stimulus-driven dynamics. It should be noted that the conditions examined here differ from those of the original framework of synaptic scaling. Our simulations pertain to a special case of homeostasis in which the target level of activity is very low (1 or 2 spikes per stimulus). Furthermore, the time frame I examined is on the scale of minutes and hours, whereas synaptic scaling is generally assumed to take place over days (Turrigiano and Nelson 2004). Under the specific conditions examined here, I observed that synaptic scaling by itself did not generate a stable dynamical state (Fig. 2). Note, however, that the oscillations might be considered irrelevant from the perspective of homeostatic plasticity, because averaged over a longer time period, the mean target activity level was achieved (Fig. 4). In the current simulations, instabilities arise for two reasons. First, as schematized in Fig. 3B, the ratio of the synaptic weights onto a neuron do not change as a result of synaptic scaling, thus preventing convergence if the solution requires a change in the synaptic ratios (Fig. 3B). Second, in a recurrent network, the input to a given neuron is nonstationary, a neuron i may find a steady-state by trial 20, and at trial 30, neuron j, which synapses onto i, may begin to fire, thus altering the behavior of i, which in turn can alter j and the network as a whole, contributing to oscillations. As shown in the toy model in Fig. 3B, these instabilities are not an issue of gain control. Specifically, the use of simulated annealing or an integral controller (van Rossum et al. 2000) will not change the ratio of the synaptic strengths.
The learning rule described in Eq. 4 takes the general form of an anti-Hebbian covariance rule in which average activity, as opposed to firing rate, is used (Dayan and Abbott 2001). By averaging long time windows, the associative flavor of the traditional covariance rules that is captured by NMDA-dependent mechanisms (Brown et al. 1990), is not present. In contrast, the induction of synaptic-scaling is not dependent on NMDA receptors (Turrigiano et al. 1998) and is viewed as a different class of learning rules (Turrigiano and Nelson 2004). Thus mechanistically, the rule presented here is likely more similar to homeostatic plasticity.
The induction mechanisms of homeostatic plasticity are poorly understood (Turrigiano and Nelson 2004). However, it is clear that presynaptic-dependent scaling would require an additional mechanism to allow for the interaction between the long-term activity levels of both the pre- and postsynaptic terminals. Such a requirement could be implemented by retrograde or orthograde messengers implicated in other forms of plasticity (e.g., Chevaleyre and Castillo 2003; Sjöström et al. 2003). Alternatively, biochemical mechanisms could be present in the postsynaptic terminal that tracks the activity levels of the presynaptic terminal by integrating glutamate signals over the course of hours/days. Under this scenario, the implementation of the rule would still be local in relation to the postsynaptic neuron.
While presynaptic-dependent scaling has not been examined experimentally, it is consistent with experiments to date, because most studies that have characterized synaptic scaling were performed with global manipulations in which all presynaptic cells would be expected to have similar Apre values. One clear prediction of the proposed learning rule is that, if only a subset of cells within a network are inactivated, postsynaptic cells would preferentially modify synapses from presynaptic cells that were not manipulated. In other words, neurons would preferentially strengthen synapses that would be more likely to increase the cell's activity level. While this type of experiment has not been explicitly examined, recent work has shown that local manipulation of postsynaptic activity does result in homeostatic synaptic plasticity (Burrone et al. 2002).
The spatial-temporal pattern of activity produced by the network is similar to that of a synfire chain (Abeles 1991; Diesmann et al. 1999). Synfire chains have been proposed as a mode of activity propagation in neural networks; however, no learning rules have described how they may emerge in a recurrent network in a self-organizing manner. While presynaptic-dependent scaling does address this issue, it is important to note that the data presented here does not account for the hypothetical millisecond precision in synfire chains (Fig. 8). Indeed, consistent with recent experimental data (Buonomano 2003), the spike variability across trials increases as a function of time.
Spike-timing–dependent plasticity (STDP) (Bi and Poo 1998; Feldman 2000; Markram et al. 1997) does not produce the type of dynamics observed here. Indeed, if STDP by itself is implemented in the preceding networks, the temporal pattern is abolished. For example, in Fig. 4B, STDP will produce a leftward shift of the spatial-temporal pattern as a result of long-term potentiation (LTP) of all the pre → post patterns. The leftward shift removes much of the temporal information and favors global synchronization, which leads to instability. However, it should be emphasized that STDP implemented together with presynaptic-dependent scaling can play a role in increasing the robustness and reliability of the responses in the presence of noise (Supplemental Fig. 1). Specifically, by strengthening the synapses in the direction of propagation (pre → post), the trajectory can become more robust in response to perturbations. The problem is that STDP must be carefully balanced with presynaptic-dependent scaling. Thus future work will have to determine if this is a biologically plausible scenario.
Calculating the average activity level
All forms of homeostatic plasticity, including the one presented here, and some models of associative plasticity (Bienenstock et al. 1982) rely on the estimation of Ai(t), which represents the average firing rate over a long time window. How neurons calculate Ai(t) is not a trivial matter. For simplicity sake, let's assume a neuron has a target activity rate of 0.1 Hz: if a stimulus is presented every 10 s, a neuron must fire once per stimulus to achieve its setpoint (as in the preceding simulations). If a stimulus is presented every 5 s, it would have to fire 0.5 times per stimulus, whereas if the intertrial interval was 60 s, it would have to fire 6 times per stimulus (even if we allow for spontaneous activity similar problems arise). Clearly such a strong dependence on the ITI is not likely to be physiological.
While there is good evidence that cells do keep track of their average level of activity (Turrigiano and Nelson 2004; Turrigiano et al. 1998), the mechanisms remain unknown. Thus it is too early to rigorously address how the dependence on intertrial interval may be solved. However, the assumption that Ai would represent activity, not over absolute time, but over states in which the animal or network is attentive or behaving, as opposed to in rest or sleep states. Specifically, the presence of a salient stimulus would result in the beginning of an integration period that would last for some predetermined period of time.
How could such selective integration work? It has been proposed that cells may track their average activity through Ca2+ sensors with long integration times (Liu et al. 1998). In vivo selective integration over some states but not others could be achieved through the same ascending neuromodulatory mechanisms that gate plasticity (e.g., Bear and Singer 1982; Kilgard 2003; Kirkwood et al. 1999). Indeed, cortical levels of acetylcholine are different during sleep, quiet wakefulness, and active wakefulness (Marrosu et al. 1995). Thus a modulator such as acetylcholine could regulate either Ca2+ influx or the downstream integration of Ca2+ signals.
One potential shortcoming of the model presented here is in relation to when more than one input is presented to the network during development. For example, what happens if on alternating trials two distinct groups of neurons are used as the input drive, and the target value of each cell remains at one? In contrast to the above simulations, there is not a unique solution to this scenario: a given neuron could fire twice on alternating trials or once on each trial. Simulations revealed that the network generally converges to one of these two solutions, and which solution was reached was dependent on initial parameters, including network connectivity and the magnitude of inhibition. However, I propose that cortical networks used for timing would indeed be triggered by a primary input and thus code time from the onset of the event whether it represented a single well-defined stimulus or a global multi-modal signal (Karmarkar and Buonomano 2003).
This model establishes how a recurrent network can implement a population clock in a self-organizing manner. While it is well established that temporal processing is a fundamental component of sensory and motor function, the neural mechanisms underlying temporal processing are not known (Ivry 1996; Mauk and Buonomano 2004). As mentioned above, it has recently been shown that subset of neurons in area HVc of the songbird implement a sparse code for time (Hahnloser et al. 2002). While the neural mechanisms by which this code is generated are not understood, this model provides a hypothesis as to how a sparse temporal code can emerge from a recurrent network.
This research was supported by National Institute of Mental Health Grant MH-60163.
The author thanks T. Blair, V. Buonomano, J. Feldman, H. Johnson, U. Karmarkar, P. Latham, and C. Marder for helpful discussions and/or reading earlier versions of this manuscript.
↵1 The Supplementary Material for this article (two movies and a figure) is available online at http://jn.physiology.org/cgi/content/full/01250.2004/DC1.
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- Copyright © 2005 by the American Physiological Society