Inferring learning rules from distributions of firing rates in cortical neurons

This is an overview of this paper.

The general question the authors try to address is whether the firing rates from a  population before and after training can be used to infer the learning rule that led to the changes in the patterns of activity.  This is is a difficult inverse problem, and requires a number of assumptions, as outlined below.The paper is motivated (and supported) by several experimental observation, including that training can lead to higher activity in a small group of cells, and reduced activity in the majority of the remainder in a population.

The authors use as motivation and analyze data from monkeys performing two different tasks: passive viewing task (monkeys view various visual stimuli and make no response), and a dimming-detection task (similar to the passive viewing task except that monkeys were required to detect and indicate a subtle decrease in luminance of the stimulus by releasing a manual lever). Neuronal responses (firing rates) to novel and familiar stimuli in inferior temporal cortex (ITC) were recorded during the tasks. Repeated presentations of an initially novel stimulus leads to a gradual decrease of responses to the stimulus in a substantial fraction of recorded neurons. The response to familiar stimuli is typically more selective,  with lower average firing rates, but higher maximum responses in putative excitatory neurons. This indicates an overall decrease in synaptic weights, except for the maximally responsive cells with increased input synaptic weights.

What type of learning rules could explain these data?  The authors assume that only recurrent weights are changed by training, and consider a rate-based plasticity rule. The firing rate of neuron i is defined by its inputs via a transfer function (f-I curve),


Training changes the recurrent synaptic weights, so that we can write


The changes in synaptic strengths lead to changes in synaptic inputs to neurons, and consequently to changes in their firing rates,


The changes in inputs can be approximately by


They make the assumption that the learning rule is a separable, i.e. the weight change depends on the  product of two functions, one depending solely on pre-, and the other solely on postsynaptic rates. Under these assumptions the dependence of the learning rule on postsynaptic firing rates is


Thus the dependence of the learning rule on the postsynaptic firing rate can be obtained from the input changes by subtracting a constant offset, and rescaling its magnitude. Note that this requires several further assumptions – importantly that everything but the change in input current is independent of i.

The assumptions that are made to deduce the learning rule are summarized in this figure.


Using the method illustrated above, they investigated the effect of visual experience in inferior temporal cortex (ITC) using neurophysiological data obtained from two different laboratories in monkeys performing two different tasks. The distributions of firing rates for novel and familiar stimuli indicated an overall negative change in input currents after learning. The authors applied the analysis outlined above separately to putative excitatory and inhibitory cells. Excitatory neurons showed negative changes when postsynaptic firing rate was low, but positive changes when it was sufficiently high. Inhibitory neurons showed negative input changes at all firing rates.

In the experimental data obtained during the passive viewing task, they further analyzed the learning effects on input currents in individual neurons. In excitatory neurons, they found diverse patterns of input changes that can be classified into three categories: neurons showing only negative changes, neurons showing negative changes for low firing rates and positive changes for high firing rates, and neurons showing only positive changes. Averaging the input change curves of excitatory neurons showing both negative and positive changes led to depression for low firing rates and potentiation for high firing rates. For neurons showing both negative and positive changes, they defined a threshold θ as the postsynaptic firing rate where input changes become positive. Denote the normalized threshold, obtained by subtracting the mean rate and dividing by the s.d. of the rate by θ′. The threshold θ was strongly correlated with both mean and s.d. of postsynaptic firing rates, but such correlation disappeared for the normalized threshold θ′. This implies that the threshold is dependent on neuronal activity, reminiscent of the BCM learning rule. The threshold observed in ITC neurons is around 1.5 s.d. above the mean firing rate, so that a majority of stimuli lead to depression while few (the ones with the strongest responses) lead to potentiation.

They next addressed whether a network model with the type of learning rule inferred from data can maintain stable learning dynamics as it is subjected to multiple novel stimuli and whether the changes of activity patterns with learning observed in the experiment can be reproduced.


When divided into subgroups by percentile ranking, most the firing rates of most groups decreased with learning in excitatory and inhibitory neurons. Only the firing rates of the excitatory neurons in the highest percentile group increased. These changes led to increased selectivity and increased sparseness for learned stimuli, in accordance with experimental data.

This paper provides an approach for deducing the learning rule from experimental data, although it requires several assumptions. Some of these assumptions may be too strong, and difficult to verify, such as the separable function of the learning rule, the rank preservation of stimuli with learning, and changes only in the recurrent weights. However, the inferred learning rule agrees with those that have been reported in other experiments. It also resembles the widely used BCM rule (2), offering further support that this approach captures at least the qualitative features of the learning rule correctly.


(1) Sukbin Lim, Jillian L McKee, Luke Woloszyn, Yali Amit,David J Freedman, David L Sheinberg & Nicolas Brunel (2015). Inferring learning rules from distributions of firing rates in cortical neurons. Nature Neuroscience 18, 1804–1810.

(2) Bienenstock, Elie L.; Leon Cooper; Paul Munro (1982). “Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex” . The Journal of Neuroscience 2 (1): 32–48.  

Comment by Krešimir Josić:  There are a number of assumptions behind this approach, that are generally explained well. However, the Gaussianity of the distribution of input currents is unclear to me. In Fig. 2b,c, this assumption is used to back out the f-I curve from responses to familiar stimuli. However, in Fig. 2f the distribution of input currents computed for familiar stimuli look non-Gaussian. How can the two be reconciled?