Delay Selection by Spike-Timing-Dependent Plasticity in Recurrent Networks of Spiking Neurons Receiving Oscillatory Inputs

Learning rules, such as spike-timing-dependent plasticity (STDP), change the structure of networks of neurons based on the firing activity. A network level understanding of these mechanisms can help infer how the brain learns patterns and processes information. Previous studies have shown that STDP selectively potentiates feed-forward connections that have specific axonal delays, and that this underlies behavioral functions such as sound localization in the auditory brainstem of the barn owl. In this study, we investigate how STDP leads to the selective potentiation of recurrent connections with different axonal and dendritic delays during oscillatory activity. We develop analytical models of learning with additive STDP in recurrent networks driven by oscillatory inputs, and support the results using simulations with leaky integrate-and-fire neurons. Our results show selective potentiation of connections with specific axonal delays, which depended on the input frequency. In addition, we demonstrate how this can lead to a network becoming selective in the amplitude of its oscillatory response to this frequency. We extend this model of axonal delay selection within a single recurrent network in two ways. First, we show the selective potentiation of connections with a range of both axonal and dendritic delays. Second, we show axonal delay selection between multiple groups receiving out-of-phase, oscillatory inputs. We discuss the application of these models to the formation and activation of neuronal ensembles or cell assemblies in the cortex, and also to missing fundamental pitch perception in the auditory brainstem.


Recurrent Correlation
As given by Equation (58) of [1], the (ordinary frequency) Fourier transform, Fg(f ) = R 1 1 g(x)e 2⇡ixf dx, of the recurrent correlation function for a network with only axonal delays is where (2) It can be considered be to make up of three components where These components are due to correlations in the inputs, spike triggering e↵ects from the inputs, and recurrent spike triggering e↵ects, respectively. The last two of these are assumed to be negligible to the learning for large numbers of inputs, M , and large numbers of neurons, N , respectively. This is the same assumption made in [1]. Because of this only the first correlation component was considered (i.e. FC(f ) ⇡ FC 1 (f )). To determine how large a network was su cient for the spike triggering components to be negligible, simulations with LIF neurons were run to observe the shape of the learned axonal delay distribution after 250s of learning. This is shown in Figure S7. For simulations it was decided that the network size would always be the same as the number of inputs (i.e. N = M ). It can be seen that as the number of neurons (and inputs) increases, the resulting delay distribution becomes a perfect cosine function. We decided that 10,000 neurons (and inputs) was su cient for simulations in this study.

Oscillatory Inputs
Input intensity functions are defined for oscillatory inputs aŝ where⌫ 0 is the mean input rate (in Hz), a is the magnitude of the oscillations (in Hz), f m is the modulation frequency of the oscillations (in Hz), andd k is the delay of the input (in seconds). Inputs within the same group have the same delay, meaning that they are in phase.
The mean input firing rate of neuron k iŝ 2 The correlation function for a pair of inputs (k and l) iŝ whered lag =d l d k , and the Fourier transform of this is If the inputs are from the same group, thend lag = 0, and sô

Homeostatic Equilibrium in a Recurrent Network
The rate of change of the recurrent axonal delay distribution iṡ where⌫(t) is the mean firing rate of the recurrent group given bȳ where ⌫ 0 is the spontaneous firing rate of the neurons,⌫ 0 is the mean firing rate of the inputs, andJ is the mean recurrent weight averaged over all axonal delays. The stable mean firing rate,⌫ ⇤ , and stable mean weight,J ⇤ , are found fromJ AssumingC W is small and that ⌫ 0 = 0, the solution to this is and, by substituting in Equation (11) from this supporting text, we have that 4 Network Response for a Single Group Given the average response is whered is the delay of the inputs. The Fourier transform of this is and by rearranging this we get For oscillatory inputs whereˆ (t) =⌫ 0 +acos(2⇡f m t) and ⇤ , the expression for the response of the network becomes where

Network Response for Two Groups
For two recurrently connected groups where the within group weights have been depressed each of the group responses are given in Equation (38) of main text. The Fourier transforms of these is and by rearranging these we get which can be approximated as This is then used to give Equation (39) in main text.

Learning Window and EPSP Kernel
It is assumed that W (u) and ✏(u) are given by and where ⌧ B > ⌧ A . From this, it can be seen that and .

Estimating the Amplitude of a Sum of Cosines
The amplitude of is unchanged under a shift in the x axis. So i a, will have the same amplitude. This can be written as where P = 1 + where the amplitude, W , is given by For the case where we have B i / X i , X < 1, and it is an infinite sum of cosines, we can estimate the square of the amplitude to the (k + 1)th order with where bxc is the floor of x.

Third-Order Covariance of Oscillatory Inputs
Similar to the second-order input covariance,