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. 2005 Apr 5;102(14):5239-44.
doi: 10.1073/pnas.0500495102. Epub 2005 Mar 28.

Generalized Bienenstock-Cooper-Munro rule for spiking neurons that maximizes information transmission

Taro Toyoizumi  1 , Jean-Pascal Pfister, Kazuyuki Aihara, Wulfram Gerstner
Affiliations

Generalized Bienenstock-Cooper-Munro rule for spiking neurons that maximizes information transmission

Taro Toyoizumi et al. Proc Natl Acad Sci U S A. .

Abstract

Maximization of information transmission by a spiking-neuron model predicts changes of synaptic connections that depend on timing of pre- and postsynaptic spikes and on the postsynaptic membrane potential. Under the assumption of Poisson firing statistics, the synaptic update rule exhibits all of the features of the Bienenstock-Cooper-Munro rule, in particular, regimes of synaptic potentiation and depression separated by a sliding threshold. Moreover, the learning rule is also applicable to the more realistic case of neuron models with refractoriness, and is sensitive to correlations between input spikes, even in the absence of presynaptic rate modulation. The learning rule is found by maximizing the mutual information between presynaptic and postsynaptic spike trains under the constraint that the postsynaptic firing rate stays close to some target firing rate. An interpretation of the synaptic update rule in terms of homeostatic synaptic processes and spike-timing-dependent plasticity is discussed.

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Figures

Fig. 2.
Fig. 2.
Relation to BCM rule. (A) The function φ(νpost) of the BCM learning rule Eq. 16 derived from our model under assumption of Poisson firing statistics of the postsynaptic neuron. A value of φ(νpost) > 0 for a given postsynaptic rate νpost means that synapses are potentiated when stimulated presynaptically. The transition from depression to potentiation occurs at a value θ that depends on the average firing rate formula image of the postsynaptic neuron (blue formula image; green formula image; red formula image). B. The threshold θ as a function of formula image for different choices of the parameter γ, i.e.,γ = 0.5 (purple); γ = 1 (black); γ = 2 (orange).
Fig. 1.
Fig. 1.
Neuron model. (A Left) Output rate νpost of the model neuron (solid line, spiking neuron model used in Figs. 3, 4, 5; green dashed line, Poisson model used in Fig. 2) as a function of presynaptic spike arrival rate at n = 100 synapses. All synapses have the same efficacy wj = 0.5, and are stimulated by independent Poisson trains at the same rate ν. (Center) Interspike interval distribution PISI of the spiking neuron model during firing at 10 (blue line), 20 (green line), or 30 Hz (red line). Firing is impossible during the absolute refractory time of τabs = 3 ms. (Right) The function g(u) used to generated action potentials (see Methods and Models for details). (B) From the first row to the fourth row (with the first row being at the top): The measure Cj that is sensitive to correlations between the state of the postsynaptic neuron and presynaptic spike arrival at synapse j, the PSPs caused by spike arrival at the same synapse j, the membrane potential u, and the postsynaptic factor Bpost of Eq. 14 as a function of time. During postsynaptic action potentials, the postsynaptic factor Bpost has marked peaks. Their amplitude and sign depend on the membrane potential at the moment of action potential firing. The coincidence measure Cj exhibits significant changes only during the duration of PSPs at synapse j.
Fig. 3.
Fig. 3.
Pattern discrimination. The first 25 synapses 1 ≤ j ≤ 25 are stimulated by Poisson input with a rate νpre = 2, 13, 25, and 40 Hz that changes each second. The remaining 75 synapses receive Poisson input at a constant rate of 20 Hz. (A Upper) Evolution of all synaptic weights as a function of time (red; strong synapses, wj ≈ 1; blue: depressed synapses, wj ≈ 0). All synapses are initialized at the same value wj = 0.1. (Lower) The evolution of the average efficacy of the 25 synapses that receive pattern-dependent input (red line) and that of the 75 other synapses (blue). Typical examples of individual traces (synapse 1: black and synapses 30: green) are given by the dashed lines. (B Upper) Evolution of the average mutual information I per bin (blue line and left scale) and of the average Kullback–Leibler distance D per bin as a function of time. Averages are calculated over segments of 1 min. (Lower) Output rate (spike count during 1 sec) as a function of pattern index before (blue bars) and after (red bars) learning.
Fig. 4.
Fig. 4.
Rate modulation. (A) Distribution of synaptic efficacies of nine postsynaptic neurons after 60 min of stimulation with identical inputs for all neurons. Synapses 1 ≤ j ≤ 40 (red symbols) received Poisson input with common rate modulation; the input at synapses 41 ≤ j ≤ 80 (blue symbols) was also rate-modulated but phase-shifted; and the input at the remaining 20 synapses was uncorrelated (green symbols). Four postsynaptic neurons (numbers 1, 3, 5, and 8) develop a spontaneous specialization for the first group of modulated input (red symbols close to the maximum efficacy of one) and five (numbers 2, 4, 6, 7, and 9) specialized for the second group. (B) Modulation of the output rate of the nine postsynaptic neurons before (Left) and after (Right) learning. Red/blue bars, neurons responding to the first/second group of input; Red/blue lines, modulation of the input of groups 1 and 2.
Fig. 5.
Fig. 5.
Spike–spike correlations. The n = 100 synapses have been separated into four groups of 25 neurons each (group A, 1 ≤ j ≤ 25; group B, 26 ≤ j ≤ 50; group C, 51 ≤ j ≤ 75; group D, 76 ≤ j ≤ 100). All synapses were stimulated at the same rate of 20 Hz. However, during the first 15 min of simulated time, neurons in groups C and D were uncorrelated, whereas the spike trains of the remaining 50 neurons (groups A and B) had correlations of amplitude c = 0.1, i.e., 10% of the spike arrival times were identical between each pair of synapses. After 15 min, correlations changed so that group A became correlated with C, whereas B and D were uncorrelated. After 45 min of simulated time, correlations stopped, but stimulation continued at the same rate. (A Upper) Evolution of all 100 weights (red, potentiated; blue, depressed). (Lower) Average mutual information per bin as a function of time. In the absence of correlations (t > 45 min), mutual information is lower than before, but the distribution of synaptic weights remains stable. (B) Nine postsynaptic neurons 1 ≤ i ≤ 9 with membrane potential ui(t) are stimulated as discussed in A and project to a readout unit with potential formula image where the sum runs over all output spikes m of all nine neurons. Mean membrane potentials are ū and h̄, respectively. The fluctuations σu =〈 (ui(t) – u ρ)2〉0.5 of the PSPs (blue line, top graph) and those of the readout potentials (σh = 〈(h(t) – h ρ)2〉0.5, green line) are correlated (Lower) with the mutual information and can serve as neuronal signal.
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