Quantitative prediction of intermittent high-frequency oscillations in neural networks with supralinear dendritic interactions

Abstract

The explanation of higher neural processes requires an understanding of the dynamics of complex, spiking neural networks. So far, modeling studies have focused on networks with linear or sublinear dendritic input summation. However, recent single-neuron experiments have demonstrated strongly supralinear dendritic enhancement of synchronous inputs. What are the implications of this amplification for networks of neurons? Here, I show numerically and analytically that such networks can generate intermittent, strong increases of activity with high-frequency oscillations; the models developed predict the shape of these events and the oscillation frequency. As an example, for the hippocampal region CA1, events with 200-Hz oscillations are predicted. I argue that these dynamics provide a plausible explanation for experimentally observed sharp-wave/ripple events. High-frequency oscillations can involve the replay of spike patterns. The models suggest that these patterns may reflect underlying network structures.

Fig. 1.

Random networks with supralinear dendritic interactions generate transiently increased activity and high-frequency oscillations with frequency 1/τ. (A) A section of the dynamics of a supralinearly coupled network with a synchronous pulse of size g′₀ = 45 generated at time t₀ = 300 ms by external stimulation. This pulse initiates a short-lived chain of propagating, enhanced synchrony. (Bottom) The spiking activity of 200 neurons. Spikes within the chain are marked red. (Middle) The network's spike rate (bin size 1 ms). (Top) The size of synchronous spike pulses within the chain. (B) The propagation of synchrony can be understood quantitatively in Markovian approximation. The chain evolution is characterized by the transition matrix (gray shaded). The dots indicate the mean response pulse sizes derived numerically (green), semianalytically (red), and analytically (blue). Between G₁ and G₂ a range with high probability of amplification exists. The gray dashed line shows the evolution of the event in A. (Inset) The dendritic modulation function σ (black line) mapping the peak excitatory postsynaptic potential (EPSP) expected from linear input summation to the effective peak EPSP.

Fig. 2.

Supralinear dendritic enhancement of inputs within a finite temporal interaction window leads to spontaneous, intermittent increases in the network activity. (A) Comparison of the depolarizations caused by several simultaneous inputs in a conventional neuron (black) and in a neuron with supralinear dendritic interactions (model 2; green). (Inset) Modulation function σ. (B and C) The dynamics of a network incorporating supralinear dendritic interactions. (B) The spike rate (bin size 1 ms) of the inhibitory (Upper) and of the excitatory (Lower) neuron population. (C) The spiking activity of 300 of the excitatory neurons. The dynamics are characterized by irregular spiking interrupted by spontaneous, intermittent increases of activity involving both the inhibitory and the excitatory neuron population. During such an event a larger fraction (about one third) of the neurons in the excitatory population sends a spike, and almost every inhibitory neuron sends usually several spikes. The event around t = 1,400 ms is depicted in Fig. 3A.

Fig. 3.

Single events in the network dynamics of model 2 consist of several subsequent pulses of highly synchronous spiking activity with temporal distance of about 5 ms. (A) An event in a random network. (Top and Middle) The spike rates (number of spikes per bin, all bin sizes 0.5 ms) of the inhibitory and of the excitatory population, respectively. (Bottom) The spiking activity of the excitatory population. (B) An event generated by a network in which only the recurrent connections of a subpopulation of the excitatory neurons allow supralinear dendritic interactions. This subpopulation (spike rate of 500 neurons; Bottom) and the inhibitory population (spike rate; Top) participate in events. Other excitatory neurons essentially do not participate (spike rate of 500 neurons; Middle). (C) An event in a network with a feed-forward structure in the excitatory population created by the presence of supralinear dendritic interactions (group sizes: 350 neurons). (Upper) The event has the usual profile (rate of the excitatory population). (Lower) The spiking activity of the excitatory population reflects the feed-forward structure in the underlying network. See main text and Fig. S4 for details.