## Abstract

This paper addresses the question how generic microcircuits of neurons in different parts of the cortex can attain and maintain different computational specializations. We show that if stochastic variations in the dynamics of local microcircuits are correlated with signals related to functional improvements of the brain (e.g. in the control of behavior), the computational operation of these microcircuits can become optimized for specific tasks such as the generation of specific periodic signals and task-dependent routing of information. Furthermore, we show that working memory can autonomously emerge through reward-modulated Hebbian learning, if needed for specific tasks. Altogether, our results suggest that reward-modulated synaptic plasticity can not only optimize the network parameters for specific computational tasks, but also initiate a functional rewiring that re-programs microcircuits, thereby generating diverse computational functions in different generic cortical microcircuits. On a more general level, this work provides a new perspective for a standard model for computations in generic cortical microcircuits (liquid computing model). It shows that the arguably most problematic assumption of this model, the postulate of a teacher that trains neural readouts through supervised learning, can be eliminated. We show that generic networks of neurons can learn numerous biologically relevant computations through trial and error.

## Introduction

It has already been shown that networks of spiking neurons with stereotypical connection probabilities can in principle support a large variety of computational tasks (Maass et al. 2002; Haeusler and Maass 2007; Haeusler et al. 2008; Buonomano and Maass 2009). In these models, termed liquid computing models, the synaptic weights of readout neurons that project to other circuits or areas are modulated by synaptic plasticity. The aim of synaptic adaptations is to approximate a desired output signal with the actual output of these readout neurons. Since these synaptic plasticity rules require knowledge of the desired output signal, this learning process is referred to as supervised learning, where a postulated supervisor or teacher supports the learning process. Furthermore, it has been shown that if such supervised learning is applied to readout neurons whose axons are also connected to other neurons within the network, such generic networks can also learn to generate periodic patterns and working memory (Jaeger and Haas 2004; Maass et al. 2007; Sussillo and Abbott 2009). These results imply that, in order to “program” the network to carry out a specific type of computation, it suffices to assign suitable values to the synaptic weights of some neurons. Furthermore, local synaptic learning rules are able to produce suitable weight settings, provided that the desired output (target output) of each readout neuron is provided at any moment during training by a teacher or supervisor. But for most concrete tasks, the assumption of such a teacher signal requires that there already exist some other neuron or network that is able to perform the desired computational task. Hence, this set-up is more suitable for duplicating a computational function, rather than for explaining how it could emerge in the first place.

We begin in this paper a new chapter in liquid computing theory, by investigating what computational properties can emerge in this approach if one eliminates all supervised learning; that is, all learning rules that require a teacher that tells a neuron how it should respond at any given time. We show that, for a large class of biologically relevant computational tasks, a teacher signal is not needed. It can be replaced by a biologically more realistic signal that assumes a high value if the average performance has recently increased and a low value otherwise. In fact, we show that several different tasks can be learned by different readouts simultaneously, based on such a relatively uninformative global feedback signal about average performance improvements. This set-up requires each neuron to explore different output values for the same network inputs. Hence, our model is not only consistent with experimentally observed stochastic features of neuronal responses (trial-to-trial variability), but requires these stochastic features for its function. In this article, for the sake of simplicity, we are discussing only rate models for generic microcircuits. Mean field models predict that sufficiently large populations of spiking neurons will behave very similar to these rate-based models (see Discussion).

The learning rule that we apply is a variation of the exploratory Hebbian (EH) rule from Legenstein et al. (2010). The EH rule is a 3-factor learning rule (Kandel and Tauc 1965a, 1965b; Bailey et al. 2000; Fiete and Seung 2006; Fremaux et al. 2010; Pawlak et al. 2010) that depends on—besides pre- and postsynaptic neural activity—a modulatory third factor. A large set of experimental data shows that neuromodulators, such as dopamine, implement such a third factor in biological networks of neurons (Reynolds et al. 2001; Reynolds and Wickens 2002; Pawlak et al. 2010), but also a modulatory signal in the form of synaptic input can influence the amplitude of the backpropagating action potential of a neuron, and thereby the learning rate of spike-timing dependent plasticity (Waters and Helmchen 2004; Sjöström and Häusser 2006). In this paper, we employ a variation of the EH rule that imposes extremely weak demands on the information provided by this third factor. Rather than assuming that it provides information regarding how much—and in which direction—the current system response deviates from some hypothetical target response, we only assume here a 2-valued global third factor M(t). This global signal informs all local plasticity mechanisms whether the system performance has recently improved. We investigate in this paper to what extent this—arguably the least informative performance-related third factor that one can conceive—suffices for installing in generic recurrent networks of neurons different, task-dependent, computational organizations. We consider 4 different types of computational tasks: In spite of the heterogeneity of these 4 computational tasks, we show that they can all be learned by the same generic neural circuit, requiring only the previously described weak information about recent performance improvement. There exists substantial evidence that neural networks of primates and other animals can carry out these tasks (see Discussion). But it has remained an open problem how biological neural networks could acquire these specific computational capabilities. In summary, the results of this work provide a new model for the emergence of diverse complex computations in biological neural systems.

1. Periodic pattern generation.

2. Learning a rule, which requires storage of specific information in working memory, and application of this stored information for online computation on complex analog input signals.

3. Context-dependent differential routing of information.

4. Nonlinear analog computations on complex input signals.

The remainder of this paper is structured as follows. We first introduce the generic microcircuit model and the reward-modulated plasticity rule used in section Materials and Methods. In Results, we then show how this network can autonomously learn to generate a complex periodic pattern. The network performance is compared with the performance of previously studied models, and the robustness of network function to various types of noise perturbations is demonstrated. We then test the same network on a completely different task that requires the emergence of 2 independent working memory stores. In a final simulation task, we show that the same network can acquire the ability to dynamically route information in a state-dependent manner. Several predictions of our model are discussed in Discussion.

## Materials and Methods

### Network Architecture

We employed a generic network model consisting of N sparsely recurrently connected neurons (with a connection probability of 0.1). We refer to these neurons as the network neurons in the following. The recurrent network model is generic in the sense that it is not designed for a particular computational function. Instead, connections within the network are randomly drawn such that network neurons are sparsely connected by excitatory and inhibitory synapses (cf. Supplementary Methods). Similar network models have been previously used to model the dynamics of recurrent biological networks of neurons (Amari 1972; Hopfield 1984; Haykin 1999; Sussillo and Abbott 2009). If necessary for the computational task, some network neurons receive in addition projections from external input streams uj(t). Specific computational functions are acquired through synaptic modification of the weights from neurons in the network to so-called readout neurons, which could represent, for example, layer 5 pyramidal cells (Maass et al. 2002; Jaeger 2003). These readout neurons can also feed back their activity into the recurrent neural network (Jaeger and Haas 2004; Maass et al. 2007). Figures 1A and 3A show the basic network topology without and with external input streams, respectively.

In our model, the state xj(t) of neuron j represents its membrane potential at the soma at time t, resulting from excitatory and inhibitory synaptic inputs (Hopfield 1984; Haykin 1999). The firing rate of the jth neuron at time t is given by

(1)
$$r_j (t) = \tan h\hbox{[}x_j\ (t)] + \xi _j^{{\rm state}}\ (t),$$
where $$\xi _j^{{\rm state}} (t)$$ models zero-mean noise on the firing rate of the neuron (see Supplementary Methods for details on noise statistics). The network dynamics is given by
(2)
$$\tau \dot x_i (t) = - x_i (t) + \lambda \sum\limits_{i = 1}^N {W_{ij}^{{\rm rec}} r_j (t) + } \sum\limits_{i = 1}^M {W_{ij}^{{\rm in}} u_j (t) + } \sum\limits_{i = 1}^L {W_{ij}^{{\rm fb}}\ z_j\ (t),}$$
where τ is the membrane time constant. The parameters $$W_{ij}^{{\rm rec}} ,\,\,W_{ij}^{{\rm in}} ,$$ and $$W_{ij}^{{\rm fb}}$$ denote the synaptic weights for recurrent connections within the network, connections from inputs to the network, and feedback connections from readout neurons to the network neurons, respectively.

Different dynamic regimes, from ordered to chaotic, can be accomplished by scaling the recurrent synaptic connections through the parameter λ. We choose a value of λ so that, as in Sussillo and Abbott (2009), the dynamics of the recurrent network tend to be in the chaotic regime prior to learning. During learning, the readout neurons drive the network activities into a nonchaotic regime via the feedback pathway. Supplementary Figure 1 illustrates the contributions of input connections, recurrent connections, and feedback connections to the membrane potential of network neurons.

We assemble the firing activities (rates) of the network neurons at time t into a column vector r(t). Assembling the synaptic weights of connections from these neurons to a readout neuron i in a corresponding column vector wi, this readout neuron i computes the function

(3)
$$\hat z_i (t) = {\bf w}_i^T {\bf r}(t) + b_i .$$

Here, the bias bi models the baseline activity (spontaneous activity) of the neuron. Our tasks do not necessitate the bias term, but we chose to adopt it for consistency with Legenstein et al. (2010). Reward-modulated Hebbian learning of readout weights requires the readout neurons to be noisy (Legenstein et al. 2010). Therefore, the output of the ith readout neuron is modeled by

(4)
$$z_i (t) = \hat z_i (t) + \xi _i (t),$$

where ξi(t) models zero-mean exploration noise on the firing rate of readout neuron i. We did not apply the tanh function for readout neurons to match the model of Sussillo and Abbott (2009). In some cases, this noisy readout signal is also fed back to the network neurons. One can apply exploration noise ξi(t) either only during learning (then $$z_i (t) = \hat z_i (t)$$ during a subsequent testing period), or during learning and testing, yielding a similar performance.

### Reward-Modulated Learning Rule

In contrast to Sussillo and Abbott (2009), who use a fully supervised online learning rule to train the network, we investigate the capabilities of a biologically more plausible reward-modulated online learning rule. The supervisor is replaced by a binary signal M(t) that communicates whether the average performance of all readouts (which are generally required to carry out different computational tasks) has recently increased. But it does not provide information on the sign and magnitude of the error, nor on which readouts contributed how much to a recent change in the performance. This single modulatory signal, which could be viewed, for example, as an abstract model for the phasic output of dopaminergic systems in the brain (Schultz et al. 1997; Schultz 2007), modulates synaptic plasticity of all synapses of all readout neurons in our simple model.

More precisely, the modulatory signal M(t) is defined by

(5)
$$M(t) = \left\{ {\matrix{ 1 & {\hbox{if}} & {P(t) \gt \bar P(t)} \cr 0 & {\hbox{if}} & {P(t) \le \bar P(t)} \cr } ,} \right.$$
where P(t) is the current performance of the system and $$\bar P(t)$$ is a low-pass filtered version of P(t) that reflects its recent performance (see Supplementary Methods for the implementation of the low-pass filter).

The current system performance P(t) is given by the sum of the mean squared errors (MSEs) of all readout neurons:

(6)
$$P(t) = - \sum\limits_{i = 1}^L {[z_i (t) - f_i (t)]^2 ,}$$
where fi(t) is the target output of readout neuron i for its assigned computational task. It is important to note that the values fi(t) of the target signals are not communicated to the neural network. This is appropriate, since their values may be unknown to a biological organism, while the existence of solutions fi(t) is assured through evolution.

We employ a variant of the EH rule proposed by Legenstein et al. (2010), where the weight change wi(t) of readout neuron i at time t is given by

(7)
$${\bf \Delta w}_i (t) = \eta (t)[z_i (t) - \bar z_i (t)]M(t){\bf r}(t).$$

Here, η(t) is a small learning rate that can either be constant, or decay over time, such that learning saturates as learning progresses. We use a decaying learning rate in all simulations reported in this paper. See Supplementary Results for additional simulation results with constant learning rates and for simulations used to determine the choice of the learning rate. Here $$\bar z_i (t)$$ is a low-pass filtered version of the noisy output zi(t) of the readout (cf. Supplementary Methods). If one assumes that $$\hat z_i (t)$$ changes only slightly within the time scale of the filter, and the noise is only weakly correlated over time, then the term $$z_i (t) - \bar z_i (t)$$ approximates the noise ξi(t). Therefore, in contrast to previously proposed rules (Fiete and Seung 2006), the rule does not need explicit information on the exploration noise, but instead estimates the noise autonomously (Legenstein et al. 2010).

This learning rule is Hebbian, since it uses the correlation of changes in the postsynaptic activity and the activity of the presynaptic neuron. It belongs to the category of 3-factor or reward-modulated Hebbian learning rules (Bailey et al. 2000; Legenstein et al. 2008; Fremaux et al. 2010; Pawlak et al. 2010), because this correlation is multiplied with the modulatory signal M(t). Since our goal was to demonstrate learning in a minimal model, we used in our simulations a variant of the original EH rule, where the modulatory signal assumes only 2 possible values. From a biological perspective, this synaptic plasticity rule only distinguishes between a high and a low modulatory signal (that could, for example, be a high or low concentration of some neuromodulator). When compared with the original formulation in Legenstein et al. (2010), where M(t) was chosen to be $$P(t) - \bar P(t)$$, this signal is much less informative. It does not communicate the magnitude of the performance change from the recent past but only whether it has improved at all. Additional simulations with the analog modulatory signal $$M(t) = P(t) - \bar P(t)$$ indicate that the performance of the 2-valued signal is comparable with the analog one; see Supplementary Results.

We point out that the exploration noise ξi(t) is the driving force of learning. Without perturbations of the readouts' output, no learning would take place.

In all our simulations, we used N = 1000 neurons in the recurrent network. These and other basic network parameters were chosen such that they correspond to the values used in Sussillo and Abbott (2009). We fixed suitable values for the other parameters that we kept for all simulations. In other words, we did not perform a parameter search in order to find the set-up with optimal performance for each individual simulation task (see Supplementary Methods for details on the parameter setting).

## Results

### Autonomous Learning of Periodic Pattern Generation

Biological neural networks produce many different types of rhythmic activities for various purposes, such as muscle activations, breathing, or locomotion. Sussillo and Abbott (2009) showed that a desired rhythmic activity can be acquired by generic recurrent neural circuits through supervised learning, where the desired output of each readout is provided at any moment during learning by a teacher or supervisor. The existence of such a teacher signal implies that some other neuron or network exists that is able to perform the task. Hence, this set-up cannot explain how the computational function could emerge in the first place. We therefore studied whether such tasks can also be learned autonomously without a teacher. We replaced the teacher signal by a modulatory signal M(t) that indicates whether the performance of the neural circuit for the considered task recently increased; see Equation (5) in Materials and Methods. The supervised learning rule used in Sussillo and Abbott (2009) was replaced by reward-modulated Hebbian learning, that is, by Hebbian synaptic plasticity that is modulated by the modulatory signal M(t); see Equation (7) in Materials and Methods. We simulated a network that receives no inputs besides the feedback projections from a single readout neuron (Fig. 1A). The task of the readout neuron was to produce a specific periodic trajectory and to repeat this periodic trajectory in a stable manner.

Figure 1.

Emergence of periodic activity through reward-modulated learning. (A) A recurrent network receives feedback from a readout neuron, but no other inputs. Circuit neurons are indicated as open circles in the gray schematic network. The readout neuron (right, with red incoming arrows indicating connections from circuit neurons) is trained to produce a specific periodic trajectory composed of 4 sinusoids with an overall period of 1 s. Red connections are subject to reward-modulated plasticity. (B) Beginning of learning (first quarter period shown). The dashed blue line indicates the onset of learning. After less than 50 ms, the readout output without exploration noise $$\hat z(t)$$ (red) approximately follows the target function f(t) (black), a prerequisite for learning with direct readout feedback (i.e., without teacher forcing). The actual feedback signal z(t) (light red) that is provided to the network includes the exploration noise ξ(t), which is the driving force of learning. (C) The beginning of the testing interval where synaptic weights remain fixed (the last second of learning and first 2 s of testing shown). The dashed green line indicates the beginning of the testing period. After a short learning time (400 s for the presented example), the readout continues to approximately produce the target function without further weight adaptation. (D) During the testing interval, the readout output shows a small drift from the target function due to the fact that the frequencies of the oscillatory components are not learned perfectly (cf. also panel H). (E and F) Emergence of a stable periodic pattern of the network state through learning. Outputs of 5 randomly chosen units of the network are shown. The dashed blue line in panel E indicates the onset of learning. While the network produces spontaneous activity before learning, a stable periodic pattern emerges shortly after the onset of learning due to the drive of the feedback loop of the readout, with a rich set of diverse activity patterns across neurons within the network. The dashed green line in panel F indicates the beginning of the testing interval. After sufficiently long learning, the stable periodic pattern keeps being produced during the testing interval. (G) Comparison of the average test error for 3 different learning rules and varying learning times. While the nonlocal RLS-based FORCE rule (green) performs best, the reward-modulated Hebbian learning rule (red) performs similar to the local LMS-based FORCE rule (blue). (H) Variability of the frequency components of the trained signal. The average amplitude spectrum across the testing interval of all simulation trials with a learning time of 400 s is shown, together with the mean and standard deviation of the peak frequencies across successful trials (MSE <0.01). Target frequency components are shown in red. While the precision of the frequency components is high in general, the small deviations lead to the drift depicted in panel D.

Figure 1.

Emergence of periodic activity through reward-modulated learning. (A) A recurrent network receives feedback from a readout neuron, but no other inputs. Circuit neurons are indicated as open circles in the gray schematic network. The readout neuron (right, with red incoming arrows indicating connections from circuit neurons) is trained to produce a specific periodic trajectory composed of 4 sinusoids with an overall period of 1 s. Red connections are subject to reward-modulated plasticity. (B) Beginning of learning (first quarter period shown). The dashed blue line indicates the onset of learning. After less than 50 ms, the readout output without exploration noise $$\hat z(t)$$ (red) approximately follows the target function f(t) (black), a prerequisite for learning with direct readout feedback (i.e., without teacher forcing). The actual feedback signal z(t) (light red) that is provided to the network includes the exploration noise ξ(t), which is the driving force of learning. (C) The beginning of the testing interval where synaptic weights remain fixed (the last second of learning and first 2 s of testing shown). The dashed green line indicates the beginning of the testing period. After a short learning time (400 s for the presented example), the readout continues to approximately produce the target function without further weight adaptation. (D) During the testing interval, the readout output shows a small drift from the target function due to the fact that the frequencies of the oscillatory components are not learned perfectly (cf. also panel H). (E and F) Emergence of a stable periodic pattern of the network state through learning. Outputs of 5 randomly chosen units of the network are shown. The dashed blue line in panel E indicates the onset of learning. While the network produces spontaneous activity before learning, a stable periodic pattern emerges shortly after the onset of learning due to the drive of the feedback loop of the readout, with a rich set of diverse activity patterns across neurons within the network. The dashed green line in panel F indicates the beginning of the testing interval. After sufficiently long learning, the stable periodic pattern keeps being produced during the testing interval. (G) Comparison of the average test error for 3 different learning rules and varying learning times. While the nonlocal RLS-based FORCE rule (green) performs best, the reward-modulated Hebbian learning rule (red) performs similar to the local LMS-based FORCE rule (blue). (H) Variability of the frequency components of the trained signal. The average amplitude spectrum across the testing interval of all simulation trials with a learning time of 400 s is shown, together with the mean and standard deviation of the peak frequencies across successful trials (MSE <0.01). Target frequency components are shown in red. While the precision of the frequency components is high in general, the small deviations lead to the drift depicted in panel D.

The rhythmic activity of the readout, which drives the network via the feedback pathway, has a strong influence on the internal network dynamics. Figure 1E,F shows a subset of 5 random units within the network at the onset of learning and at the transition from learning to testing, respectively. Before the system starts with the learning process, the network exhibits chaotic dynamics and produces rich spontaneous activity. Shortly after the onset of learning, a stable periodic pattern emerges due to the driving force of the feedback loop (panel E). This stable periodic pattern persists during the testing interval when there is no further weight adaptation (panel F). A certain level of chaoticity—which we regulate by the parameter λ that scales the weights of the recurrent network—is necessary for an accurate performance of the system. Initial chaotic dynamics are needed because the network has to initially produce sufficiently rich dynamics to properly generate the target function. On the other hand, if the chaoticity exceeds a certain level, the drive from the feedback loop is too weak to drive the network dynamics into a stable regime (Supplementary Fig. 2D; see also Supplementary Fig. 1 for the contributions of input connections, recurrent connections, and feedback connections to the membrane potential of network neurons). This is consistent with the results of Sussillo and Abbott (2009).

To investigate the learning time needed such that the network reliably autonomously reproduces the oscillatory pattern, we conducted 50 independent simulation trials with different learning times from 10 to 400 s. Each simulation trial consisted of a learning interval of varying length and a subsequent testing interval of 500 s. Moreover, in order to evaluate whether the elimination of the teacher leads to a significant decrease in performance, we conducted the same simulations with systems employing 2 different FORCE learning rules and compared the performance of the systems (cf. Supplementary Methods for a brief description of the FORCE learning rules). The supervised FORCE learning rules have previously been tested for readout training on similar tasks (Sussillo and Abbott 2009). Figure 1G shows the result of this comparison. The recursive least squares (RLS)-based FORCE rule (green) performs best, leading to a good approximation of the target signal after learning for as few as 10 s (which corresponds to 10 oscillation cycles of the target pattern). This is not surprising since the RLS-based rule uses nonlocal information about the correlations between all pairs of inputs to the readout to adapt the individual synapses. However, this approach seems to be problematic from the point of biological plausibility. The reward-modulated Hebbian learning rule (red) performs similarly to the local least mean squares (LMS)-based FORCE rule (blue) that still requires full knowledge of the desired output signal. With both of these local learning rules, the network needs to learn for approximately 100 s before a good performance level is reached. The desired trajectory was stably produced until the end of the testing interval in the majority of simulation trials (cf. Supplementary Fig. 2C). A comparison of the performance distributions across simulation trials showed that the performance of our learning rule did not differ significantly from the performance of the LMS FORCE rule for learning times of >100 s. The performance of our learning rule remained approximately constant across learning times >150 s (i.e., there was no significant difference between the performances at any 2 training times >150 s; nonparametric Wilcoxon rank-sum tests, significance level: 5%). For the comparison between the LMS FORCE rule and the EH rule, learning rates were independently set to good values that were obtained by a brute force search (see Supplementary Results for the choice of the learning rates for the 2 rules).

Biological networks of neurons have to be able to operate under the influence of substantial amounts of noise and other perturbances. Therefore, we have tested the noise robustness of the model. The results show that the system, which has learned without a supervisor, is indeed robust to noisy perturbations of the network state and the readout output as well as to long clamping of the readout output after learning. Additionally, substantial amounts of noise can also be applied to network neurons during learning (Supplementary Results).

### Necessary Conditions

To investigate the limits of this approach, we tested the system behavior under various conditions. We performed simulations where we varied some of the system parameters and the properties of the target pattern. Three factors were varied concurrently, leading to a total of 45 parameter settings for which the network was tested (Fig. 2). Specifically, we investigated the influence of the following network and target pattern properties on the ability of the network to generate a periodic target pattern:

1. Frequency components of the target pattern (columns in Fig. 2A);

2. Update interval of the weights and the modulatory signal M(t) (rows in Fig. 2A); and

3. Time constant of the exploration noise (x-axis of histograms in Fig. 2A).

Figure 2.

Necessary conditions for the emergence of periodic activity through reward-modulated Hebbian learning. (A) Simulations with varying target pattern frequency components (horizontally) and weight update intervals (vertically). Each panel shows the distribution of the MSE across 50 simulation trials per value of the exploration noise autocorrelation time constant (1–5 ms). Letters in brackets associate performance levels with the example readout outputs in panel B. (B) Representative example traces (black) of the readout activity that show the system behavior in simulation trials with varying performance during the testing period and target output (red). The panel indices (a–d) are being indicated by the indices with arrows in panel A. Note that the time scale of panel (d) is different from that of the other panels.

Figure 2.

Necessary conditions for the emergence of periodic activity through reward-modulated Hebbian learning. (A) Simulations with varying target pattern frequency components (horizontally) and weight update intervals (vertically). Each panel shows the distribution of the MSE across 50 simulation trials per value of the exploration noise autocorrelation time constant (1–5 ms). Letters in brackets associate performance levels with the example readout outputs in panel B. (B) Representative example traces (black) of the readout activity that show the system behavior in simulation trials with varying performance during the testing period and target output (red). The panel indices (a–d) are being indicated by the indices with arrows in panel A. Note that the time scale of panel (d) is different from that of the other panels.

The frequency components and therefore the rate of change of the target pattern are an important factor for the difficulty of the task. If the target signal changes too fast, the readout is not able to adapt its output quickly enough to follow the target pattern during learning. Additionally, in order to assure that temporal averages are estimated with sufficient accuracy in the EH rule, the noise-free readout activation should change slowly on the time scale of these averages (see Legenstein et al. 2010 for details). Each column of Figure 2A corresponds to simulations with a set of given frequency components. The periodic target pattern in the above simulations had a length of 1 s with frequency components between 1 and 4 Hz (middle column). We performed simulations where this target pattern was scaled to lengths of 0.5 and 2 s, corresponding to frequencies of 2–8 Hz (right column) and 0.5–2 Hz (left column), respectively. As expected, learning works well if frequency components are sufficiently slow (up to 4 Hz in our simulations). Performance for patterns with higher frequency components can be strongly improved if the network time constant τ is reduced (cf. Supplementary Fig. 3).

It was assumed in the above simulations that the modulatory signal M(t) is provided at every time step. Based on this signal, the synaptic weights to the readout neurons were updated as frequently. The frequency of the weight update and update of M(t) is a crucial parameter because weight updates have to be sufficiently fast compared with the system evolution in order to be able to adapt to the target trajectory quickly after the onset of learning and to follow the target trajectory during the learning process. If the system is not driven into the expected regime by the feedback signal, the corresponding transformation from the network state to the desired output trajectory cannot be learned properly by the readout. Each row in Figure 2A corresponds to simulations with a given update interval: 1 ms (top row; as in our initial simulations), 5 ms (middle row), and 10 ms (bottom row). The results indicate that longer update intervals are still possible if frequency components in the target pattern are sufficiently slow. As expected, learning of fast frequency components is not possible if update intervals are long.

In the theoretical justification of the EH rule, it is assumed that the exploration noise in the readout neurons is not (or at least only weakly) correlated over time (Legenstein et al. 2010). This may not always be justified in biological readout neurons. Temporally correlated exploration noise is expected to lead to worse performance since the readout neuron can explore fewer settings in a given time interval, which again leads to slower adaptation of the readout output. We tested the stability of the system against temporal correlations in the exploration noise by performing simulations with exploration noise that is temporally correlated with time constants τnoise between 1 and 5 ms (Supplementary Methods). In Figure 2A, the x-axis of each of the 2-dimensional histograms is dedicated to the exploration noise time constant. As expected, low noise correlations are beneficial for the learning process. With increasing frequencies in the target pattern, the system is likely to fail to reach appropriate performance levels if the exploration noise is temporally correlated (upper right panel). For additional simulation results with different target patterns, see Supplementary Results.

### Autonomous Learning of a Computational Rule and of the Working Memory that it Requires

Many cognitive operations of the brain require a working memory, where specific task-relevant information is stored for intervals up to a few seconds. Neuronal correlates of working memory have been observed, for example, in single neuron recordings from the prefrontal cortex of macaque monkeys during visual working memory in delayed matching-to-sample tasks (Fuster and Alexander 1971; Goldman-Rakic 1995; Miller et al. 1996; Bernacchia et al. 2011). In these experiments, it was observed that prefrontal cortex neurons hold information of previously observed stimuli by a persistent increase or decrease of their firing rates for a time interval in the range of seconds.

Figure 3A shows the set-up for this task. Figure 3B shows a representative example of network performance after 500 s of learning. The last 30 s of a subsequent 500-s testing interval are presented. The firing rates of the 2 readouts correctly change depending on the associated inputs. Hence, the system learned the correct behavior for both readouts based on a single modulatory signal, which indicated only whether the combined performance of both readouts recently improved. We performed 50 independent simulation trials and calculated the percentage of time in which the readouts are in the wrong state (output value closer to 0.5 rather than −0.5, or vice versa) during a 500-s testing interval. The average fraction of time in the wrong state is 4.57 ± 0.73% of the testing time, with an even lower median fraction of 2.9% (based on 50 simulation trials, average proportion over both readouts per simulation trial).

Figure 3.

Emergence of task-specific working memory in a generic neural network through reward-modulated Hebbian learning. (A) In this task, 2 readouts are trained by adapting their weights (red and purple arrows in panel A) using a common binary modulatory signal. Each readout zi(t) is trained to produce a memory trace [red and purple traces in panel B, the black trace represents the target function fi(t)] by changing its firing rate depending on the activity of the 2 input streams it is associated with. Input neurons are indicated by colored open circles on the left. If the associated “on” input (uon,i(t), green and orange inputs in panel A, and green and orange traces in panel B) is briefly activated, the readout is trained to switch to a high firing rate. If the associated “off” input (uoff,i(t), dark and light blue inputs in panel A, and dark and light blue traces in panel B) is activated, it is trained to switch to a low firing rate. (B) The last 30 s of a testing period after 500 s of learning are shown for readout z1(t) in the upper part (together with the inputs uon,1(t) and uoff,1(t) that determined its desired value). Traces of z2(t), uon,2(t), and uoff,2(t) are shown in the lower part. Both readouts nicely produce the desired memory traces.

Figure 3.

Emergence of task-specific working memory in a generic neural network through reward-modulated Hebbian learning. (A) In this task, 2 readouts are trained by adapting their weights (red and purple arrows in panel A) using a common binary modulatory signal. Each readout zi(t) is trained to produce a memory trace [red and purple traces in panel B, the black trace represents the target function fi(t)] by changing its firing rate depending on the activity of the 2 input streams it is associated with. Input neurons are indicated by colored open circles on the left. If the associated “on” input (uon,i(t), green and orange inputs in panel A, and green and orange traces in panel B) is briefly activated, the readout is trained to switch to a high firing rate. If the associated “off” input (uoff,i(t), dark and light blue inputs in panel A, and dark and light blue traces in panel B) is activated, it is trained to switch to a low firing rate. (B) The last 30 s of a testing period after 500 s of learning are shown for readout z1(t) in the upper part (together with the inputs uon,1(t) and uoff,1(t) that determined its desired value). Traces of z2(t), uon,2(t), and uoff,2(t) are shown in the lower part. Both readouts nicely produce the desired memory traces.

To analyze the behavior of the network in this task, we performed a principal component analysis (PCA) of the network activity vectors r(t) before learning and during the test epoch. Figure 4 shows the network trajectories in this task projected onto the first 3 principal components of the test trajectories before learning (panel A), early during learning (panel B), and during testing (panel C). As expected, before learning starts, the network dynamics are not restricted to specific attractor subregions (panel A) and network dynamics are high dimensional (panel D). Early in the learning process, the feedback from the readout neurons—which is constantly adapted by the EH learning rule—already constrains the network to a low-dimensional subspace (panel D). We identified 4 subareas that are visited with likelihoods that depend on the momentary readout state (panel B). This behavior is preserved after learning (panel C,D). This indicates that the adaptation of the readout weights autonomously generates attractor regions in the activity landscape of the network to implement the working memory necessary for this task. Pulses at the network inputs move the activity state of the network to the corresponding region.

Figure 4.

Working memory induce low-dimensional network dynamics. (A) First 3 principal components of the network trajectories in the working memory task before learning (PCA performed on network trajectories before learning). The network exhibits rich dynamics with no signs of attracting subregions. (B and C) First 3 principal components (PCA performed on network trajectories after learning) of the network trajectories during the early learning phase (B) and after learning (C). The state of the 2 readouts is indicated in color in panels B and C (red: z1(t) < −0.3, z2(t) < −0.3; blue: z1(t) < −0.3, z2(t) > 0.3; cyan: z1(t) > 0.3, z2(t) > 0.3; green: z1(t) > 0.3, z2(t) < −0.3; gray otherwise). Clearly, the network dynamics visits 4 subspaces depending on the actual readout state during and after learning. (D) Percentage of variance that can be explained by the first 1–10 principal components (#PCs) before learning (brown crosses), during learning (yellow circles), and during testing (black “x”). During testing and learning, the network dynamics are low dimensional since the first 2 principal components are able to explain most of the variance of the network dynamics.

Figure 4.

Working memory induce low-dimensional network dynamics. (A) First 3 principal components of the network trajectories in the working memory task before learning (PCA performed on network trajectories before learning). The network exhibits rich dynamics with no signs of attracting subregions. (B and C) First 3 principal components (PCA performed on network trajectories after learning) of the network trajectories during the early learning phase (B) and after learning (C). The state of the 2 readouts is indicated in color in panels B and C (red: z1(t) < −0.3, z2(t) < −0.3; blue: z1(t) < −0.3, z2(t) > 0.3; cyan: z1(t) > 0.3, z2(t) > 0.3; green: z1(t) > 0.3, z2(t) < −0.3; gray otherwise). Clearly, the network dynamics visits 4 subspaces depending on the actual readout state during and after learning. (D) Percentage of variance that can be explained by the first 1–10 principal components (#PCs) before learning (brown crosses), during learning (yellow circles), and during testing (black “x”). During testing and learning, the network dynamics are low dimensional since the first 2 principal components are able to explain most of the variance of the network dynamics.

In summary, these simulations show that a generic neural circuit can learn a computational rule and simultaneously hold information about recently observed inputs for at least several seconds. This learning took place without any supervision, just from information about recent changes in global system performance.

Figure 5.

Figure 5.

### Analysis of Emerging Computational Mechanisms

Figure 6.

Figure 6.

How did the pulse input u1 influence the working memory? Neurons with large ccMem had on average no preference for input channel u1 which initiates the persistent activity of readout z1. Instead, we found a number of neurons in the network that were correlated with the pulse input u1; see Figure 6F (and analogous neurons for u2, not shown). These correlations ccu1 were not particularly strong. But as shown in Figure 6G for the most strongly correlated neuron, these neurons had peak activities whenever u1 was active (compare with panel E), resulting from strong synaptic inputs from u1 (Supplementary Fig. 5E). The direct influence of such neurons on neurons with large ccMem was presumably weak, as their weights to these neurons were small on average (Supplementary Fig. 5F). We thus hypothesized that the readout utilized these neurons to switch itself and therefore the whole attractor. Surprisingly, we found only a slight correlation between ccu1 of a network neuron and its weight to the readout neuron (Supplementary Fig. 5G). To further elaborate on this point, we performed control simulations where we set all weights in the recurrent network to zero. We found that no attractor was attained. Instead, the memory readout mimicked u1 and u2. This shows that some disynaptic connections—from the switching inputs to the network, and then directly to the readout—substantially participated in switching the readout.

In summary, this brief analysis shows that surprisingly complex computational mechanisms can be induced in a generic recurrent circuit by adapting the synaptic efficacies to readout neurons with a reward-modulated Hebbian learning rule.

## Discussion

We have shown that heterogeneous and specialized computational functions can emerge through synaptic plasticity in generic sparsely connected recurrent networks of neurons. Furthermore, different computational structures can emerge simultaneously, even without an instructive teacher signal that tells each neuron what it should compute. Finally, we have shown that neural circuits need to receive very little information about the target output of each readout neuron: It suffices if they all receive a single global signal, which informs them whether their combined average performance has recently improved, or not. Arguably, this is the least informative performance-related signal that one can possibly conceive. We conjecture that if one reduces the information content of this global signal even further, then goal-directed learning is no longer possible.

A common feature of the 4 computational tasks considered in this paper is that there exists substantial evidence that neural networks of primates and other animals can carry out these tasks. Primates can learn to generate an immense variety of periodic movements, but it is not known how these capabilities are acquired and stored. Our results imply that, in principle, no specialized genetically encoded neural networks are required for this. This is of interest, because to the best of our knowledge no evidence for the existence of the latter has been found in primates. Furthermore, there exists a large body of experimental evidence that primates (Rodriguez and Paule 2009) and rodents (Rich and Shapiro 2009; Durstewitz et al. 2010) can learn rules for behaving that depend on specific cues, in a way that is likely to be rewarded. To accomplish such behavior, they must be able to keep relevant cues in working memory for extended periods of time. But it has remained an open problem how such rules are learned, and how working memory is implemented in neural networks of the brain. One remarkable recent experimental study (Bernacchia et al. 2011) suggests that working memory is implemented through heterogeneous neural subpopulations with different temporal responses, in a manner similar to that which emerges through reward-modulated learning in our study. A number of researchers have postulated that context- and task-dependent routing of information is required in the brain for language understanding (Dominey et al. 2006), and for many other tasks where abstract knowledge needs to be applied for processing sensory input streams (Olshausen et al. 1993). Previous models for flexible routing of information in networks of neurons (Anderson and Van Essen 1987; Olshausen et al. 1995; von der Malsburg 2003; Zylberberg et al. 2010) required specially constructed neural networks. Recently, Vogels and Abbott (2009) proposed a model that is based on balanced excitation and inhibition. Additionally, several models have been introduced where synchronous activity is utilized for information routing in neural networks (Crick and Koch 1990; Salinas and Sejnowski 2001; Fries 2005; Akam and Kullmann 2010). Our work proposes a new model that relies neither on assumptions about synchronous network activity nor explicitly on balanced synaptic input. According to our model, generic networks of neurons can learn to route information as needed for specific computational tasks. Finally, it had already been shown in Maass et al. (2002) that specific analog computations on complex input signals can be learned by the readouts from a generic recurrent network of neurons through supervised learning. We show here that this learning also succeeds with a biologically more realistic feedback signal that does not require a teacher.

Some specific conditions on the structure and dynamics of networks of neurons have to be met in order to make this learning result possible. One is that the output of neural circuits needs to exhibit trial-to-trial variability (i.e., stochasticity). Without that, no exploration of possibly better network specialization is possible. Another condition is that neural circuits need to have some basic features, which enable them to acquire virtually arbitrary computational specializations by just changing the synaptic weights of a few neurons. The underlying theory (Maass et al. 2007) guarantees only that this can be achieved if readout neurons can compute arbitrary continuous functions (without memory). However, if the neural circuit provides sufficiently rich generic nonlinear preprocessing (see the kernel property of Legenstein and Maass 2007), linear readouts tend to suffice (Maass et al. 2002; Legenstein and Maass 2007). Finally, for some of the desired computations (those that require a nonfading memory), it is necessary that readout neurons whose synaptic inputs are subject to synaptic plasticity also project their output back into the circuit.

In this paper, we considered only rate models for generic cortical microcircuits. Similar network models have been previously used to model the dynamics of recurrent biological networks of neurons (Amari 1972; Hopfield 1984; Haykin 1999; Sussillo and Abbott 2009). It has been shown in computer simulations that some related computational tasks can be learned by networks of integrate-and-fire neurons through supervised learning (Maass et al. 2007). This indicates that the general set-up is in principle compatible with networks of spiking neurons, although the network size will have to be increased considerably for comparable performance. Reward-modulated Hebbian learning rules similar to the EH rule used in this paper have also been formulated for spiking neurons (Fiete and Seung 2006). Such spike-based learning rules could be employed by spiking readout neurons. One major difference between the EH rule [Equation (7)] and related reward-modulated Hebbian learning rules is the estimate of the exploration noise ξi(t) with the help of a low-pass filtered version $$\bar z_i (t)$$ of the current firing rate zi(t). The low-pass filtered version of the firing activity could be implemented in a biological neuron through its molecular machinery. For example, the intracellular concentration of Ca2+ of a neuron is related to its recent firing history. This is also demonstrated by the fact that many pyramidal neurons exhibit spike frequency adaptation. Our model also predicts that information about recent increases or decreases in the firing rate is available to the molecular machinery that implements synaptic plasticity. But how this could be implemented is a wide open question, as are most questions regarding the molecular mechanisms that implement synaptic plasticity, and their exact time courses.

### Experimentally Testable Predictions of Our Model

A primary prediction of our model is that no drastic differences in the structure of cortical microcircuits that perform different types of computations (periodic pattern generations, working memory, switchable routing of information, and nonlinear online computations) should be expected. Furthermore, our model predicts that a large number of different brain areas can learn to carry out these computations. In fact, several studies indicate that network function is only partially genetically predetermined, for example, in networks that generate periodic patterns for locomotion (Marder and Goaillard 2006) and in the auditory cortex (von Melchner et al. 2000). Other predictions of our model concern the organization of learning, more precisely the information provided by modulatory signals that gate synaptic plasticity.

Many studies indicate that the frontal cortex contains neurons that are sensitive to errors (Ridderinkhof et al. 2004) as well as neurons that track past or future performance (Hasegawa et al. 2000). In traditional error-based learning approaches involving multiple readouts, each readout unit was supervised individually by providing it with its exact error (or even with its target value). According to our model, it is not necessary that an individual modulatory signal is provided to each readout. Instead, we use the same modulatory signal for all readouts, indicating only whether the collective performance has increased due to random noise perturbations. Consider, for example, a task where the overall error is given by the combined error in several subtasks. Our model suggests that such tasks can in principle be learned, even if only the overall error but not the individual error signals can be extracted from the sensory information. The fact that animals are able to learn motor tasks that demand the coordination of many muscle activations or motor synergies indicates that this is indeed possible.

Movement-related rhythmic activity patterns related to jaw and tongue movements have been found in the primary motor cortex of primates (Yao et al. 2002), which has been shown to be involved in learning of fine motor skills (Molina-Luna et al. 2009). Traces of persistent memory, reflected in sustained firing activity of single neurons in response to specific visual stimuli, have been recorded in the prefrontal cortex (Fuster and Alexander 1971; Goldman-Rakic 1995; Miller et al. 1996). Primary motor cortex and prefrontal cortex receive input projections from midbrain dopaminergic neurons, and the release of dopamine from such projections has been related to the expression of synaptic plasticity in this area (Molina-Luna et al. 2009; Hosp et al. 2011). Therefore, the entrainment of such movement-related activity patterns in the primary motor cortex and persistent memory traces in the prefrontal cortex may also be guided by modulatory input from midbrain neuromodulatory signals. Assuming that the proposed tasks are performed using such a learning mechanism, our model predicts that the synaptic adaptations that keep the desired trajectories stable during learning depend on the availability of global signals, such as specific neuromodulators. Without the presence of such signals, adaptation would not be possible. This is consistent with studies showing that both working memory performance and motor skill learning are impaired if the dopaminergic system of the brain is degenerated, as in patients with Parkinson's disease (Durstewitz and Seamans 2002; Doyon 2008; Molina-Luna et al. 2009), and also with studies showing that working memory performance is impaired if dopaminergic input to the prefrontal cortex is blocked (Durstewitz and Seamans 2002). Moreover, our simulation results are also consistent with results indicating that dopaminergic signaling in the primary motor cortex is involved in learning new motor skills, but not in executing a previously learned skill (Molina-Luna et al. 2009; Hosp et al. 2011).

### Conclusion

In summary, we have shown how diverse computational functions, such as periodic pattern generation, memory-dependent computations, and state-dependent routing of information, could be attained and maintained by generic cortical microcircuits via a biologically plausible 3-factor learning rule. It suffices that local synaptic learning rules receive a global modulatory signal that transmits some minimal information about global changes in task performance. Our results suggest that neuronal variability plays a crucial role in this learning process. It enables generic networks of neurons to learn important computational tasks without any supervisor or teacher (as postulated in previous work on the liquid computing model), simply through trial and error.

## Supplementary Material

Supplementary material can be found at: http://www.cercor.oxfordjournals.org/

## Funding

This work was supported by the ORGANIC project (#FP7-231267), the Brain-i-Nets project (#FP7-243914), and the AMARSi project (#FP7-248311) of the European Union.

## Notes

We also thank 2 anonymous reviewers for fruitful suggestions and Andrew Whitford for his helpful comments on the manuscript. Conflict of Interest: None declared.

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The brain's router: a cortical network model of serial processing in the primate brain
PLoS Comput Biol
,
2010
, vol.
6
pg.
e1000765