Resonance with subthreshold oscillatory drive organizes activity and optimizes learning in neural networks

Abstract

Network oscillations across and within brain areas are critical for learning and performance of memory tasks. While a large amount of work has focused on the generation of neural oscillations, their effect on neuronal populations' spiking activity and information encoding is less known. Here, we use computational modeling to demonstrate that a shift in resonance responses can interact with oscillating input to ensure that networks of neurons properly encode new information represented in external inputs to the weights of recurrent synaptic connections. Using a neuronal network model, we find that due to an input current-dependent shift in their resonance response, individual neurons in a network will arrange their phases of firing to represent varying strengths of their respective inputs. As networks encode information, neurons fire more synchronously, and this effect limits the extent to which further "learning" (in the form of changes in synaptic strength) can occur. We also demonstrate that sequential patterns of neuronal firing can be accurately stored in the network; these sequences are later reproduced without external input (in the context of subthreshold oscillations) in both the forward and reverse directions (as has been observed following learning in vivo). To test whether a similar mechanism could act in vivo, we show that periodic stimulation of hippocampal neurons coordinates network activity and functional connectivity in a frequency-dependent manner. We conclude that resonance with subthreshold oscillations provides a plausible network-level mechanism to accurately encode and retrieve information without overstrengthening connections between neurons.

Keywords: forward replay; oscillations; reverse replay; sequence learning; spiking resonance.

Fig. 1.

Input-dependent resonance shift allows for selectively activating subsets of neurons. (A) Model neurons receive three types of input. External input is DC, which varies in magnitude with neuron identity, represented by the color mapped arrow. All neurons receive an identical oscillating input, represented by the sine wave. Additionally neurons receive the synaptic inputs from neighboring neurons according to network connectivity and synaptic weights. (B) The input-dependent resonance shift manifests as a broadening of the resonance curve with increasing excitation of the neurons. (C) Broadening of the resonance curve also occurs for changes in synaptic weights, which provides for selective activation of subsets of neurons based on synaptic coupling. Dashed lines show the frequencies corresponding to the raster plots in D–F, which show the divergent activation for frequencies between 12 and 16 Hz. Error bars, $\pm$SEM.

Fig. 2.

Resonating networks learn by mapping input patterns to synaptic weights. (A) Raster plots show the relationship between the phase of firing and the external input to the neuron. Black lines show the trace of the oscillating input, and the color of the rasters shows the DC input to the given neuron. Neurons are sorted by their input rank. Subpanels in A correspond to before DC input distribution is applied (pre-), with DC input distribution (input), after learning has saturated (post-), after learning/no DC distribution (replay 1), and after a second period of learning with no DC distribution (replay 2). (B) The relationship between firing phase and DC input varies between negatively, positively, and not correlated for different epochs of the simulation. Data are averaged over 10 cycles of the oscillation. Error bars = $\pm$SEM. (C) Transitioning from the input pattern-depending firing phases to synchronous firing is gradual. Lines trace the firing phase of 12 neurons with varying input magnitudes across time. The horizontal bars above indicate when the external input and learning are present (white, input but no learning; black, learning and input; gray, no input and no learning).

Fig. 3.

Replay of stored pattern occurs independent of neuronal model and frequency band. (A) Input-induced pattern of firing phase for a network of Ks neurons driven with a 6 Hz oscillation. (B) Reversal of pattern during replay after learning for a network of Ks neurons driven by a 6 Hz oscillation. (C) Reversed pattern replayed by a network of HH neurons at 60 Hz. All raster plots include spike from 10 cycles of the oscillation, and the color of a neuron’s raster indicates the magnitude of DC input it gets in a pattern. (D) Firing phase versus DC input relations for the three above cases (red $\to$ Ks neuron before learning, green $\to$ Ks neuron replay, blue $\to$ HH neuron replay). Error bars = $\pm$SEM.

Fig. 4.

Learning saturates naturally after input pattern is completely mapped to synapses. Saturation of learning reliably occurs given that the learning rate is high enough for the given time. Both maximum (A) and mean (B) synaptic weight saturate. Line color indicates network learning rate. (C) The majority of synaptic change occurs early during the learning period and then gradually decreases to zeros. (D) Final mean synapse strength and time until learning saturates depends on the spread of the input distribution. Error bars = $\pm$SEM.

Fig. 5.

Input pattern maps to both synaptic inputs and outputs. After learning, input strength (black) is anticorrelated with input magnitude of a neuron in the pattern, and output strength (red) is correlated. Error bars = $\pm$SEM.

Fig. 6.

Sequential activation of network subgroups leads to phase precession through time. (A) Networks were sequentially activated by a slowly varying depolarizing current delivered to subsets of neurons (solid lines; color indicates group). Spiking activity of each group is represented by the raster plots of different colors. Neurons are sorted on the y axis based on their lattice location, such that neurons closer on the y axis are more likely to be connected. (B and C) At the transition between the activation of two groups, phase order changes so that the neurons receiving the highest activation always fire at an earlier phase, leading to a phase precession through time. This is shown (B) in relation to the activation of the groups and (C) with reference to the oscillation. (D) The activation versus phase relationship shows that neurons fire earlier and with less variability with more depolarizing input. Error bars = $\pm$SEM.

Fig. 7.

Learned sequences can be replayed in both the forward and reverse directions. After a sequence is learned, (A) reverse replay occurs when a network is driven by an oscillation and (B) forward when the network is driven by noise. Raster plots follow the same organization as Fig. 6. (C and D) The firing relationships between groups is stable across cycles and different from groups without learning. (E) Synaptic connections between groups encode the sequence direction between groups, while weakening the reverse direction. (F) The directionality of intergroup connections emerges gradually after repeated sequence presentations, with increasing variations between groups. Error bars = $\pm$SEM.

Fig. 8.

Resonating networks have organized functional structure over a narrow frequency band. Theta band resonance leads to a highly organized functional network structure. In simulated networks, spike–LFP coherence (A), mean AMD z score (B), and functional network stability (C) all dramatically increase between 4 and 10 Hz. This effect is robust to noise, which is indicated by line color. (D) In vivo optogenetic stimulation of hippocampal PV+ neurons lead to similar increases in spike–LFP coherence and functional network stability at these frequencies. Error bars = $\pm$SEM.

Fig. 9.

The model proposes a mechanism for the generation of reverse replay. Reverse replay due to how an input pattern imposes a phase procession of neuron firing due with respect to the oscillation. As the network learns the pattern, inputs to weakly excited neurons are strengthened while those to highly excited neurons are weakened. When the pattern is removed, inputs from synaptic connections dominate, and the reverse mapping of synaptic weights leads to reverse reactivation.