The chronotron: a neuron that learns to fire temporally precise spike patterns

Abstract

In many cases, neurons process information carried by the precise timings of spikes. Here we show how neurons can learn to generate specific temporally precise output spikes in response to input patterns of spikes having precise timings, thus processing and memorizing information that is entirely temporally coded, both as input and as output. We introduce two new supervised learning rules for spiking neurons with temporal coding of information (chronotrons), one that provides high memory capacity (E-learning), and one that has a higher biological plausibility (I-learning). With I-learning, the neuron learns to fire the target spike trains through synaptic changes that are proportional to the synaptic currents at the timings of real and target output spikes. We study these learning rules in computer simulations where we train integrate-and-fire neurons. Both learning rules allow neurons to fire at the desired timings, with sub-millisecond precision. We show how chronotrons can learn to classify their inputs, by firing identical, temporally precise spike trains for different inputs belonging to the same class. When the input is noisy, the classification also leads to noise reduction. We compute lower bounds for the memory capacity of chronotrons and explore the influence of various parameters on chronotrons' performance. The chronotrons can model neurons that encode information in the time of the first spike relative to the onset of salient stimuli or neurons in oscillatory networks that encode information in the phases of spikes relative to the background oscillation. Our results show that firing one spike per cycle optimizes memory capacity in neurons encoding information in the phase of firing relative to a background rhythm.

Figure 1. A graphical illustration of the chronotron problem for a neuron with 2 synapses.

(A) The dynamics of the membrane potential . The numbered arrows indicate the timings when the membrane potential reaches the firing threshold and spikes are fired. (B) The dynamics of the two components of . (C) The trajectory of . Spikes are generated when the trajectory reaches the spike-generating hyperplane, which is here a line. The chronotron problem is solved by adjusting the location of the spike-generating hyperplane, through changes in , such that the timings of the fired spikes are the target ones. The numbered arrows indicate the generation of spikes at the times when the spike-generating line is reached. The neuron has .

Figure 2. A graphical illustration of the chronotron problem for a neuron with 2 synapses (continued).

As in Fig. 1, but for other values of , resulted through the application of E-learning, starting from the situation in Fig. 1, and having as a target the generation of one spike at 75 ms. Left: during learning. Right: after learning converged.

Figure 3. A graphical illustration of the chronotron problem for a neuron with 3 synapses.

(A) The dynamics of the membrane potential . The numbered arrows indicate the timings when the membrane potential reaches the firing threshold and spikes are fired. (B) The dynamics of the three components of . (C) The trajectory of . Spikes are generated when the trajectory reaches the spike-generating hyperplane, which is here the black plane. The numbered arrows indicate the generation of spikes at the timings when the spike-generating hyperplane is reached. The neuron has .

Figure 4. The error landscape for a neuron with two synapses and the descent on this landscape during learning.

The neuron receives several input spikes on each synapse, the same as in Figs. 1 and 2, and has to fire one spike at a predefined target timing, the same as in Fig. 2. (A), (B) A contour plot of the VP and E distances between the actual spike train and the target spike train as a function of the values of the synaptic efficacies. The thick lines correspond to discontinuities of the distances. (A) VP distance. (B) E distance. (C), (D), (E) The dynamics of the synaptic efficacies according to the learning rules. The black lines represent actual trajectories of the synaptic efficacies. The vectors represent synaptic changes. The green line corresponds to the values of the synaptic efficacies for which the output corresponds to the target spike train. (C) E-learning. (D) I-learning. (E) ReSuMe.

Figure 5. A graphical illustration of the plastic changes implied by the learning rules.

The graphs show the spike timings and, for one synapse, the dynamics of the synaptic current , the normalized PSP and the synaptic changes implied by the two learning rules. It is considered that one input spike arrives at this synapse at . The synaptic changes are shown to be localized temporally along the events that cause them; the actual application of the synaptic changes can be delayed with respect to these events. (A) One independent target spike and no actual spike. (B) A pair of matching target and actual spikes, the actual one following the target one. (C) One independent actual spike and no target spike. (D) A pair of matching target and actual spikes, the target one following the actual one.

Figure 6. Learning of a mapping between one input pattern and one output spike train.

The trained neuron receives inputs from 500 neurons. The spike trains received from these neurons form the input pattern. Each input spike train consists of one spike within the 200 ms of a trial, generated at a random timing having an uniform distribution within the trial. The target output spike train consists of spikes at 50, 100 and 150 ms. (A) Part of the input pattern and the output spike train of the trained neuron, corresponding to this input, before learning. Only some of the 500 input spike trains are illustrated. (B) The synaptic efficacies change according to E-learning, such that the trained neuron's output reproduces the target spike train. Left: The output spike train during learning. Right: The VP distance between the actual and the target output spike train, during learning. The target output is reproduced after less than 15 epochs (presentations of the input pattern). (C) The VP distance between the actual and the target output spike train during learning, for E-learning and I-learning: averages and standard deviations over 10,000 realizations of the same experiment. Each realization uses different, random input spike trains and initial values of the synaptic efficacies.

Figure 7. Learning of a mapping between 10 input patterns, with and without jitter, and one output spike train.

Left: The VP distance between the actual and the target output spike train. Center: The timing difference between matching spikes and the target spikes. Right: The probability that the fired spikes matched the target ones. The graphs represent averages and standard deviations over input patterns and over 10,000 realizations. (A)–(D): Evolution during learning, as a function of the learning epoch. (A), (B): No jitter. (C), (D): A gaussian jitter with an amplitude of 5 ms is added to each presentation of the input patterns. (E), (F): Values after 400 learning epochs, as a function of the amplitude of the input jitter. (A), (C), (E): E-learning. (B), (D), (F): I-learning. The inputs and the trial length are as in Fig. 6. The target output spike train consists of one spike at 100 ms.

Figure 8. The distribution of the synaptic efficacies, before and after learning, for the experiments presented in Fig. 7 .

(A) Before learning. (B)–(E) After 400 learning epochs. (B), (D) E-learning. (C), (E) I-learning. (B), (C) No jitter. (D), (E) A gaussian jitter with an amplitude of 5 ms is applied to the inputs.

Figure 9. The performance of the chronotron learning rules for a classification problem.

The input patterns are classified into 3 classes. (A)–(C) The average minimum number of epochs required for correct learning is displayed as a function of the load , for various values of the number of input synapses . Note the scale differences. (A) E-learning. (B) I-learning. (C) ReSuMe. (D) The maximum load for which correct learning can be achieved (the capacity ), as a function of . E-learning has a much better performance than I-learning or ReSuMe. For E-learning, simulations for higher were not performed because of the high computational cost, due to the high capacity resulted through this learning rule. Averages were computed over 500 realizations with different, random initial conditions.

Figure 10. The dependence on the number of categories of the performance of E-learning for a classification problem.

(A) The average minimum number of epochs required for correct learning, as a function of the load , for various numbers of categories . Regardless of , the points fall on the same curve. (B) The maximum load for which correct learning is achieved (the capacity ), as a function of the number of categories . The shaded area represents the uncertainty due to the fact that the load can vary only discretely, in steps of , for a particular . The capacity is approximately constant for all .

Figure 11. The dependence of chronotron performance on the number of output spikes per trial.

The neuron had to learn to have the same output for all inputs, using E-learning. The output consisted of output spikes, placed at , for . (A) The maximum load (the capacity ) as a function of the number of output spikes . (B) The number of learning epochs required for correct learning as a function of the number of output spikes , for various loads . (C) The number of learning epochs required for correct learning as a function of load, for various numbers of output spikes . Best performance was achieved for a single output spike per trial.

Figure 12. The dependence of chronotron performance on the firing rate of the inputs.

The inputs were generated using a Gamma process having a normalized average period (the average period over the trial length) (Methods). (A) The maximum load (the capacity ) as a function of the normalized average period . (B) The number of learning epochs required for correct learning as a function of the normalized average period , for various loads . (C) The number of learning epochs required for correct learning as a function of load , for various values of the normalized average period . Best capacity was achieved for values of around 1, i.e. a single input spike per trial, for each synapse, on average, while fastest learning was achieved for around 0.5.

Figure 13. The dependence of chronotron performance on the probability that input synapses receive no spikes.

At the beginning of the experiment, each input spike train was set up as either one spike generated at a random timing or, with a probability , of no spikes. Input patterns did not change during learning. (A) The maximum load (the capacity ) as a function of the no firing probability . (B) The number of learning epochs required for correct learning as a function of the no firing probability , for various loads . (C) The number of learning epochs required for correct learning as a function of load , for various values of the no firing probability . Best capacity was achieved for values of less or equal to 0.1, while fastest learning was achieved when there was no input with no spikes. For large there are not enough input spikes to drive the neuron and, as expected, performance drops.

Figure 14. The dependence of chronotron performance on the timing of the output spike and on the initial state of the membrane potential.

The neuron had to learn to have the same output for all inputs. The output was one spike at a given timing . At the beginning of each trial, the membrane potential was either set to , as in the other experiments (stable initial state), or was generated randomly, with a uniform distribution, between 0 and (random initial state). (A) The maximum load (the capacity ) as a function of the timing of the output spike . (B) The number of learning epochs required for correct learning as a function of the timing of the output spike , for various loads . (C) , as a reference for comparing the effect on learning of the initial conditions, as a function of the timing of the output spike . For this setup, the capacity and the learning time for reaching the correct output, for stable initial state, does not depend on if it is larger than about 40 ms. Because of the exponential decay of the membrane potential of the chronotron with a time constant , the effect of the random initial state of the membrane potential on the chronotron's performance, as a function of the output spike timing , becomes insignificant at about , similarly to , as .

Figure 15. The dependence of chronotron performance on trial length.

(A) The maximum load (the capacity ) as a function of the trial length . (B) The number of learning epochs required for correct learning as a function of the trial length , for various loads . (C) The number of learning epochs required for correct learning as a function of load , for various values of the trial length . Best performance was achieved for ms (the relevant parameter is , ).

Figure 16. The dependence of chronotron performance on the reset potential.

(A) The maximum load (the capacity ) as a function of the reset potential . (B) The number of learning epochs required for correct learning as a function of the reset potential , for various loads . (C) The number of learning epochs required for correct learning as a function of load , for various values of the reset potential . The performance does not depend on the reset potential if it is lower than half of the firing threshold, mV.

Figure 17. The performance of learning rules when their parameters were optimized for fast learning for, ().

(A) The number of learning epochs required for correct learning as a function of the load , for . Correct learning was not achieved for I-learning and ReSuMe for larger than 0.03. (B) The number of learning epochs required for correct learning as a function of the number of input synapses . Correct learning was not achieved for I-learning for nor larger than 6,000. Averages and standard deviations over 500 realizations. The arrows indicate the conditions for which the parameters were optimized.

Figure 18. The dependence of chronotron performance on when synapses are updated during simulations.

The number of learning epochs required for correct learning as a function of the load , for various methods of applying the synaptic changes according to the learning rules: batch updating (synapses are changed at the end of each batch of trials, each one corresponding to one of the input patterns); trial updating (synapses are changed at the end of each trial); online updating (synapses are changed after each target or actual postsynaptic spike — for I-learning only). (A) E-learning. (B) I-learning.

Figure 19. The kernels used in the simulation of the integrate-and-fire neuron.

(A) The kernel. (B), (C) The kernel. In (B) there is no postsynaptic spike. In (C), a postsynaptic spike is fired at ms. A presynaptic spike is received at .

Figure 20. The distribution of the number of input spikes per trial, for inputs generated using a Gamma process, as in Fig. 12 .

(A) The normalized average period is . (B) . (C) . (D) . (E) .