Oscillation-Induced Signal Transmission and Gating in Neural Circuits

Reliable signal transmission constitutes a key requirement for neural circuit function. The propagation of synchronous pulse packets through recurrent circuits is hypothesized to be one robust form of signal transmission and has been extensively studied in computational and theoretical works. Yet, although external or internally generated oscillations are ubiquitous across neural systems, their influence on such signal propagation is unclear. Here we systematically investigate the impact of oscillations on propagating synchrony. We find that for standard, additive couplings and a net excitatory effect of oscillations, robust propagation of synchrony is enabled in less prominent feed-forward structures than in systems without oscillations. In the presence of non-additive coupling (as mediated by fast dendritic spikes), even balanced oscillatory inputs may enable robust propagation. Here, emerging resonances create complex locking patterns between oscillations and spike synchrony. Interestingly, these resonances make the circuits capable of selecting specific pathways for signal transmission. Oscillations may thus promote reliable transmission and, in co-action with dendritic nonlinearities, provide a mechanism for information processing by selectively gating and routing of signals. Our results are of particular interest for the interpretation of sharp wave/ripple complexes in the hippocampus, where previously learned spike patterns are replayed in conjunction with global high-frequency oscillations. We suggest that the oscillations may serve to stabilize the replay.

Synchronous spiking of neurons in some layer causes a synchronous input to the neurons of the next layer.
In the presence of oscillations of suitable frequency, e.g., ν s ≈ ν nat , this input may be supported by inputs from the external oscillations. Then the total excitatory input I = I e + I c (S1.1) is the sum of inputs arising from external oscillations, I e , and from the preceding layer, I c . In networks with non-additive coupling, the spiking probability p sp due to a synchronous input I below the dendritic threshold Θ b is typically much smaller than due to a suprathreshold input (cf. Figure 2a). We thus assume that only neurons that receive a suprathreshold input (I > Θ b ) generate a spike with xed probability p * , i.e., (S1.2) Thus, neurons process synchronous signals like simple threshold units, i.e., they generate no response for subthreshold inputs and a xed response for suprathreshold inputs. For clarity of presentation, we assume that the ring probability p * is xed. In general it might be reduced by inhibitory input, but the extension is straightforward and leads to similar results (cf. also [41]).
The timing of somatic spikes initiated by dendritic spikes is highly precise, i.e., the temporal distribution of somatic spikes triggered by dendritic ones is very narrow (cf. Figure 3a), in the sub-millisecond range (cf. also [29]). In particular, the jitter in time is typically much smaller than the dendritic integration window ∆T s . This let us assume that a synchronous pulse packet in one layer causes synchronous spiking within a time interval smaller than ∆T s in the next layer and so on.
In the following we calculate the probability density function f I (I) for the total excitatory input I to the neurons of a given layer conditioned on (i) the number of synchronously spiking neurons in the previous layer, g in , and (ii) the amplitude of external oscillations, N e . Then, the average number of synchronously spiking neurons in the considered layer is (S1.4) First we consider the input from the previous layer. Given the random topology of the FFN, the probability that a neuron receives exactly k (out of the maximal number g in ) inputs is binomially distributed, (S1.5) For a suciently large number g in of neurons participating in the synchronous pulses, we can approximate the binomial distribution (S1.5) by a Gaussian distribution and thus the excitatory synchronous input follows and standard deviation (S1.8) Likewise, the number of excitatory inputs l each neuron receives within one oscillation period from the external (virtual) neuron population is binomially distributed, Gaussian distribution with standard deviation σ s . We assume that propagation of synchrony in the FFN and the external oscillations are in-phase. Then for σ s > 0 the fraction (S1.13) The sum of the inputs I e and I c is then also approximately Gaussian distributed, I = I e + I c ∼ N µ, σ 2 , (S1.14) with mean µ = µ e + µ c and variance σ 2 = σ 2 e + σ 2 c , i.e., µ = ε ext p N e e p ext ex + ε c g in p ex (S1.15) and σ = ε ext (S1.16) Using the distribution (S1.14) of I allows us to specify the iterated map for the average size of a synchronous pulse according to Equation (S1.4), where the size of the initial pulse packet g in appears as argument of µ and σ (see Equations S1.15 and S1.16). The xed points G * = g out = g in of Equation (S1.17) determine the stability of the propagation of a synchronous pulse. With increasing coupling strength two xed points emerge via a tangent bifurcation (Figure S1.1a; cf. also Figure 2b,c), and external oscillations have a similar eect ( Figure S1.1b). This transition enables robust propagation of synchrony, and the external oscillations thus reduce the critical connection strength ε * NL (i.e., the minimal coupling strength for which robust signal propagation is possible).
For a given network setup, N e = N * e species the minimal size of the external oscillation which enables stable propagation of synchrony. It can be found by numerically determining the bifurcation point of Equation (S1.17). Additionally, one can derive a scaling law for N * e based on two observations: 1. In the absence of external oscillations (N e = 0), the position of the bifurcation point of Equation (S1.17) depends on the coupling strength ε c and the dendritic threshold Θ b only via the quotient which is the number of spikes from the preceding layer that are needed to elicit a dendritic spike. Equation (S1.17) reads For a given network setup, the connection probability p ex , group size ω and spiking probability p * (which is determined by the ground state and the parameters of the dendritic spike) are xed. Thus the bifurcation point where the xed points G * 1 = G * 2 = g out = g in appear by a tangent bifurcation, depends solely on κ (the only unknown quantity). Consequently, there is some κ * = κ specifying this bifurcation point, i.e., the transition point from non-propagating to propagating regime depends just on the number of spikes necessary to elicit a dendritic spike. The actual value κ * can be found either by numerical simulation of the system, numerical solution of Equation (S1.19) or by the analytical methods introduced in [30, 31].

The main inuence of external oscillatory inputs is
an eective reduction of the dendritic threshold Θ b , such that the properties of the system described above can be approximated by a network without external oscillatory input, but with a reduced dendritic threshold Θ e b < Θ b : In the setups considered the additional oscillatory input contributes to the generation of dendritic spikes, but the main contribution arises from the input arriving from the previous layer (the signal to be propagated), i.e., µ e < µ c . Moreover, the feed-forward connections ε c are enhanced compared to the remaining excitatory couplings, ε ext p < ε c . Thus the total variation of the input σ = σ 2 e + σ 2 c (cf. Equation S1.16) is dominated by the contribution σ 2 c of the input from the previous layer, ε ext p µ e 1 − p ext ex < ε p µ c (1 − p ex ) (S1.20) σ 2 e < σ 2 c . (S1.21) In particular for ε ext p ε c the contribution of the external inputs to the total variation of the input becomes negligible, i.e., σ 2 e σ 2 c , and the argument of the error function in Equation (S1.17) simplies to where we dened the eective dendritic threshold (S1.23) The above observations indicate that the bifurcation point is found for some constant (S1.24) such that the minimal size of the external oscillations N * e , which enables propagation of synchrony, is given by (using Equations S1.13, S1.11 and S1.10) (S1.25) Equation (S1.25) indicates that N * e changes linearly with the coupling strength ε c (cf. Figure 4 and 5). Further it is inversely proportional to the coupling strength between the external oscillatory population and the neurons of the FFN, N * e ∝ 1/ε ext p , and the dependence of N * e on the temporal spread σ s of the external oscillations is determined by the prefactor 1/p ∆T s .
The above results are derived for isolated FFNs. However, we show and discuss in Supporting Material Text S2 that the results hold in good approximation also for FFNs that are part of recurrent networks.