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A learning theory for reward-modulated spike-timing-dependent plasticity with application to biofeedback.

Legenstein R, Pecevski D, Maass W - PLoS Comput. Biol. (2008)

Bottom Line: This article provides tools for an analytic treatment of reward-modulated STDP, which allows us to predict under which conditions reward-modulated STDP will achieve a desired learning effect.These analytical results imply that neurons can learn through reward-modulated STDP to classify not only spatial but also temporal firing patterns of presynaptic neurons.Our model for this experiment relies on a combination of reward-modulated STDP with variable spontaneous firing activity.

View Article: PubMed Central - PubMed

Affiliation: Institute for Theoretical Computer Science, Graz University of Technology, Graz, Austria.

ABSTRACT
Reward-modulated spike-timing-dependent plasticity (STDP) has recently emerged as a candidate for a learning rule that could explain how behaviorally relevant adaptive changes in complex networks of spiking neurons could be achieved in a self-organizing manner through local synaptic plasticity. However, the capabilities and limitations of this learning rule could so far only be tested through computer simulations. This article provides tools for an analytic treatment of reward-modulated STDP, which allows us to predict under which conditions reward-modulated STDP will achieve a desired learning effect. These analytical results imply that neurons can learn through reward-modulated STDP to classify not only spatial but also temporal firing patterns of presynaptic neurons. They also can learn to respond to specific presynaptic firing patterns with particular spike patterns. Finally, the resulting learning theory predicts that even difficult credit-assignment problems, where it is very hard to tell which synaptic weights should be modified in order to increase the global reward for the system, can be solved in a self-organizing manner through reward-modulated STDP. This yields an explanation for a fundamental experimental result on biofeedback in monkeys by Fetz and Baker. In this experiment monkeys were rewarded for increasing the firing rate of a particular neuron in the cortex and were able to solve this extremely difficult credit assignment problem. Our model for this experiment relies on a combination of reward-modulated STDP with variable spontaneous firing activity. Hence it also provides a possible functional explanation for trial-to-trial variability, which is characteristic for cortical networks of neurons but has no analogue in currently existing artificial computing systems. In addition our model demonstrates that reward-modulated STDP can be applied to all synapses in a large recurrent neural network without endangering the stability of the network dynamics.

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Setup of the model for the experiment by Fetz and Baker [17].(A) Schema of the model: The activity of a single neuron in the circuit                            determines the amount of reward delivered to all synapses between                            excitatory neurons in the circuit. (B) The reward signal                                d(t) in response to a spike train                            (shown at the top) of the arbitrarily selected neuron (which was                            selected from a recurrently connected circuit consisting of 4000                            neurons). The level of the reward signal                            d(t) follows the firing rate of the                            spike train. (C) The eligibility function                                fc(s) (black curve,                            left axis), the reward kernel                                εr(s) delayed                            by 200 ms (red curve, right axis), and the product of these two                            functions (blue curve, right axis) as used in our computer experiment.                            The integral of                                    fc(s+dr)εr(s)                            is positive, as required according to Equation 10 in order to achieve a                            positive learning rate for the synapses to the selected neuron.
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pcbi-1000180-g003: Setup of the model for the experiment by Fetz and Baker [17].(A) Schema of the model: The activity of a single neuron in the circuit determines the amount of reward delivered to all synapses between excitatory neurons in the circuit. (B) The reward signal d(t) in response to a spike train (shown at the top) of the arbitrarily selected neuron (which was selected from a recurrently connected circuit consisting of 4000 neurons). The level of the reward signal d(t) follows the firing rate of the spike train. (C) The eligibility function fc(s) (black curve, left axis), the reward kernel εr(s) delayed by 200 ms (red curve, right axis), and the product of these two functions (blue curve, right axis) as used in our computer experiment. The integral of fc(s+dr)εr(s) is positive, as required according to Equation 10 in order to achieve a positive learning rate for the synapses to the selected neuron.

Mentions: We show that this phenomenon can in principle be explained by reward-modulated STDP. In order to do that, we define a model for the experiment which allows us to formulate an equation for the reward signal d(t). This enables us to calculate synaptic weight changes for this particular scenario. We consider as model a recurrent neural circuit where the spiking activity of one neuron k is recorded by the experimenter (Experiments where two neurons are recorded and reinforced were also reported in [17]. We tested this case in computer simulations (see Figure 2) but did not treat it explicitly in our theoretical analysis). We assume that in the monkey brain a reward signal d(t) is produced which depends on the visual feedback (through an illuminated meter, whose pointer deflection was dependent on the current firing rate of the randomly selected neuron k) as well as previously received liquid rewards, and that this signal d(t) is delivered to all synapses in large areas of the brain. We can formalize this scenario by defining a reward signal which depends on the spike rate of the arbitrarily selected neuron k (see Figure 3A and 3B). More precisely, a reward pulse of shape εr(r) (the reward kernel) is produced with some delay dr every time the neuron k produces an action potential(9)Note that d(t) = h(t)−h̅ is defined in Equation 1 as a signal with zero mean. In order to satisfy this constraint, we assume that the reward kernel εr has zero mass, i.e., . For the analysis, we use the linear Poisson neuron model described in Methods. The mean weight change for synapses to the reinforced neuron k is then approximately (see Methods)(10)This equation describes STDP with a learning rate proportional to . The outcome of the learning session will strongly depend on this integral and thus on the form of the reward kernel εr. In order to reinforce high firing rates of the reinforced neuron we have chosen a reward kernel with a positive bump in the first few hundred milliseconds, and a long negative tail afterwards. Figure 3C shows the functions fc and εr that were used in our computer model, as well as the product of these two functions. One sees that the integral over the product is positive and according to Equation 10 the synapses to the reinforced neuron are subject to STDP. This does not guarantee an increase of the firing rate of the reinforced neuron. Instead, the changes of neuronal firing will depend on the statistics of the inputs. In particular, the weights of synapses to neuron k will not increase if that neuron does not fire spontaneously. For uncorrelated Poisson input spike trains of equal rate, the firing rate of a neuron trained by STDP stabilizes at some value which depends on the input rate (see [24],[25]). However, in comparison to the low spontaneous firing rates observed in the biofeedback experiment [17], the stable firing rate under STDP can be much higher, allowing for a significant rate increase. It was shown in [17] that also low firing rates of a single neuron can be reinforced. In order to model this, we have chosen a reward kernel with a negative bump in the first few hundred milliseconds, and a long positive tail afterwards, i.e. we inverted the kernel used above to obtain a negative integral . According to Equation 10 this leads to anti-STDP where not only inputs to the reinforced neuron which have low correlations with the output are depressed (because of the negative integral of the learning window), but also those which are causally correlated with the output. This leads to a quick firing rate decrease at the reinforced neuron.


A learning theory for reward-modulated spike-timing-dependent plasticity with application to biofeedback.

Legenstein R, Pecevski D, Maass W - PLoS Comput. Biol. (2008)

Setup of the model for the experiment by Fetz and Baker [17].(A) Schema of the model: The activity of a single neuron in the circuit                            determines the amount of reward delivered to all synapses between                            excitatory neurons in the circuit. (B) The reward signal                                d(t) in response to a spike train                            (shown at the top) of the arbitrarily selected neuron (which was                            selected from a recurrently connected circuit consisting of 4000                            neurons). The level of the reward signal                            d(t) follows the firing rate of the                            spike train. (C) The eligibility function                                fc(s) (black curve,                            left axis), the reward kernel                                εr(s) delayed                            by 200 ms (red curve, right axis), and the product of these two                            functions (blue curve, right axis) as used in our computer experiment.                            The integral of                                    fc(s+dr)εr(s)                            is positive, as required according to Equation 10 in order to achieve a                            positive learning rate for the synapses to the selected neuron.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2543108&req=5

pcbi-1000180-g003: Setup of the model for the experiment by Fetz and Baker [17].(A) Schema of the model: The activity of a single neuron in the circuit determines the amount of reward delivered to all synapses between excitatory neurons in the circuit. (B) The reward signal d(t) in response to a spike train (shown at the top) of the arbitrarily selected neuron (which was selected from a recurrently connected circuit consisting of 4000 neurons). The level of the reward signal d(t) follows the firing rate of the spike train. (C) The eligibility function fc(s) (black curve, left axis), the reward kernel εr(s) delayed by 200 ms (red curve, right axis), and the product of these two functions (blue curve, right axis) as used in our computer experiment. The integral of fc(s+dr)εr(s) is positive, as required according to Equation 10 in order to achieve a positive learning rate for the synapses to the selected neuron.
Mentions: We show that this phenomenon can in principle be explained by reward-modulated STDP. In order to do that, we define a model for the experiment which allows us to formulate an equation for the reward signal d(t). This enables us to calculate synaptic weight changes for this particular scenario. We consider as model a recurrent neural circuit where the spiking activity of one neuron k is recorded by the experimenter (Experiments where two neurons are recorded and reinforced were also reported in [17]. We tested this case in computer simulations (see Figure 2) but did not treat it explicitly in our theoretical analysis). We assume that in the monkey brain a reward signal d(t) is produced which depends on the visual feedback (through an illuminated meter, whose pointer deflection was dependent on the current firing rate of the randomly selected neuron k) as well as previously received liquid rewards, and that this signal d(t) is delivered to all synapses in large areas of the brain. We can formalize this scenario by defining a reward signal which depends on the spike rate of the arbitrarily selected neuron k (see Figure 3A and 3B). More precisely, a reward pulse of shape εr(r) (the reward kernel) is produced with some delay dr every time the neuron k produces an action potential(9)Note that d(t) = h(t)−h̅ is defined in Equation 1 as a signal with zero mean. In order to satisfy this constraint, we assume that the reward kernel εr has zero mass, i.e., . For the analysis, we use the linear Poisson neuron model described in Methods. The mean weight change for synapses to the reinforced neuron k is then approximately (see Methods)(10)This equation describes STDP with a learning rate proportional to . The outcome of the learning session will strongly depend on this integral and thus on the form of the reward kernel εr. In order to reinforce high firing rates of the reinforced neuron we have chosen a reward kernel with a positive bump in the first few hundred milliseconds, and a long negative tail afterwards. Figure 3C shows the functions fc and εr that were used in our computer model, as well as the product of these two functions. One sees that the integral over the product is positive and according to Equation 10 the synapses to the reinforced neuron are subject to STDP. This does not guarantee an increase of the firing rate of the reinforced neuron. Instead, the changes of neuronal firing will depend on the statistics of the inputs. In particular, the weights of synapses to neuron k will not increase if that neuron does not fire spontaneously. For uncorrelated Poisson input spike trains of equal rate, the firing rate of a neuron trained by STDP stabilizes at some value which depends on the input rate (see [24],[25]). However, in comparison to the low spontaneous firing rates observed in the biofeedback experiment [17], the stable firing rate under STDP can be much higher, allowing for a significant rate increase. It was shown in [17] that also low firing rates of a single neuron can be reinforced. In order to model this, we have chosen a reward kernel with a negative bump in the first few hundred milliseconds, and a long positive tail afterwards, i.e. we inverted the kernel used above to obtain a negative integral . According to Equation 10 this leads to anti-STDP where not only inputs to the reinforced neuron which have low correlations with the output are depressed (because of the negative integral of the learning window), but also those which are causally correlated with the output. This leads to a quick firing rate decrease at the reinforced neuron.

Bottom Line: This article provides tools for an analytic treatment of reward-modulated STDP, which allows us to predict under which conditions reward-modulated STDP will achieve a desired learning effect.These analytical results imply that neurons can learn through reward-modulated STDP to classify not only spatial but also temporal firing patterns of presynaptic neurons.Our model for this experiment relies on a combination of reward-modulated STDP with variable spontaneous firing activity.

View Article: PubMed Central - PubMed

Affiliation: Institute for Theoretical Computer Science, Graz University of Technology, Graz, Austria.

ABSTRACT
Reward-modulated spike-timing-dependent plasticity (STDP) has recently emerged as a candidate for a learning rule that could explain how behaviorally relevant adaptive changes in complex networks of spiking neurons could be achieved in a self-organizing manner through local synaptic plasticity. However, the capabilities and limitations of this learning rule could so far only be tested through computer simulations. This article provides tools for an analytic treatment of reward-modulated STDP, which allows us to predict under which conditions reward-modulated STDP will achieve a desired learning effect. These analytical results imply that neurons can learn through reward-modulated STDP to classify not only spatial but also temporal firing patterns of presynaptic neurons. They also can learn to respond to specific presynaptic firing patterns with particular spike patterns. Finally, the resulting learning theory predicts that even difficult credit-assignment problems, where it is very hard to tell which synaptic weights should be modified in order to increase the global reward for the system, can be solved in a self-organizing manner through reward-modulated STDP. This yields an explanation for a fundamental experimental result on biofeedback in monkeys by Fetz and Baker. In this experiment monkeys were rewarded for increasing the firing rate of a particular neuron in the cortex and were able to solve this extremely difficult credit assignment problem. Our model for this experiment relies on a combination of reward-modulated STDP with variable spontaneous firing activity. Hence it also provides a possible functional explanation for trial-to-trial variability, which is characteristic for cortical networks of neurons but has no analogue in currently existing artificial computing systems. In addition our model demonstrates that reward-modulated STDP can be applied to all synapses in a large recurrent neural network without endangering the stability of the network dynamics.

Show MeSH
Related in: MedlinePlus