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Conceived and designed the experiments: JL LS KO. Performed the experiments: JL MA. Analyzed the data: JL KO. Contributed reagents/materials/analysis tools: JL LS KO. Wrote the paper: JL MA LS KO.

The authors have declared that no competing interests exist.

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.

Synchronization of neuronal spiking in the brain is related to cognitive functions, such as perception, attention, and memory. It is therefore important to determine which properties of neurons influence their collective behavior in a network and to understand how. A prominent feature of many cortical neurons is spike frequency adaptation, which is caused by slow transmembrane currents. We investigated how these adaptation currents affect the synchronization tendency of coupled model neurons. Using the efficient adaptive exponential integrate-and-fire (aEIF) model and a biophysically detailed neuron model for validation, we found that increased adaptation currents promote synchronization of coupled excitatory neurons at lower spike frequencies, as long as the conduction delays between the neurons are negligible. Inhibitory neurons on the other hand synchronize in presence of conduction delays, with or without adaptation currents. Our results emphasize the utility of the aEIF model for computational studies of neuronal network dynamics. We conclude that adaptation currents provide a mechanism to generate low frequency oscillations in local populations of excitatory neurons, while faster rhythms seem to be caused by inhibition rather than excitation.

Synchronized oscillating neural activity has been shown to be involved in a variety of cognitive functions

The phase response curve (PRC) provides a powerful tool to study neuronal synchronization

In recent years substantial efforts have been exerted to develop single neuron models of reduced complexity that can reproduce a large repertoire of observed neuronal behavior, while being computationally less demanding and, more importantly, easier to understand and analyze than detailed biophysical models. Two-dimensional variants of the leaky integrate-and-fire neuron model have been proposed which take into consideration an adaptation mechanism that is spike triggered

Because of subthreshold and spike-triggered contributions to the adaptation current, the aEIF model exhibits a rich dynamical structure

The aEIF model consists of two differential equations and a reset condition,

The dynamics of the model relevant to our study is outlined as follows. When the input current

We selected realistic values for the model parameters (

A–C: Membrane potential

In order to compare the effects of adaptation in the aEIF model with those of

The dynamics of interest is described below. Starting from a resting state, as

We used parameter values as in

We considered networks of

We simulated the aEIF and Traub neuron networks, respectively, taking

We measured the degree of spike synchronization in the simulated networks using averaged pairwise cross-correlations between the neurons

In order to quantify the degree of phase locking of neurons in the network we applied the mean phase coherence measure

The PRC can be obtained (experimentally or in simulations) by delivering small perturbations to the membrane potential of a neuron oscillating with period

For Traub model trajectories, the peak of the action potential is identified with phase

The PRCs presented in this study were calculated using the adjoint method. For validation purposes, we also simulated a number of PRCs by directly applying small perturbations to the membrane potential

In the limit of weak synaptic interaction, which guarantees that a perturbed spiking trajectory remains close to the attracting (unperturbed) trajectory

We first examine the effects of the adaptation components

In

A,B: PRCs associated with adaptation parameters as in

To provide an intuitive explanation for the effects of adaptation on the PRC, we show the vector fields,

We next investigate how the changes in PRCs caused by either adaptation component are affected by the spike frequency. Bifurcation currents, rheobase currents and corresponding frequencies, in dependence of

A,B: Rheobase current (solid black), SN and AH bifurcation currents

In this section, we examine how the changes in phase response properties due to adaptation affects phase locking of coupled pairs of periodically spiking aEIF neurons. Specifically, we first analyze how the shape of the PRC determines the fixed points of eq. (23) and their stability, and then show how the modifications of the PRC mediated by the adaptation components

In case of identical cell pairs and symmetric synaptic strengths,

First, consider a synaptic current with infinitely fast rise and decay. In this case we use a positive (or negative)

A synaptic current with finite rise and decay times causes an additional rightwards shift and a smoothing of the interaction function. The stability of the fixed point

A: PRC of an aEIF neuron (top) spiking at

First, consider pairs of identical aEIF neurons with the PRCs shown in

Next, we consider pairs that are coupled through synaptic currents

Stable (solid black) and unstable (dashed grey) phase locked states of pairs of aEIF neurons spiking at

We next investigate how phase locked states of excitatory and inhibitory pairs are affected by synaptic currents that involve conduction delays, considering the PRC of a neuron without adaptation, and two PRCs that represent adaptation induced by either

Stable (solid black) and unstable (dashed grey) phase locked states of aEIF pairs without adaptation,

Spike times (solid bars) of two neurons oscillating with a small phase difference

In the following we analyze phase locking of neuronal pairs with unequal synaptic peak conductances

A–C: Change of phase difference

In order to examine how the behavior of pairs of coupled phase neurons relates to networks of spiking neurons, we performed numerical simulations of networks of oscillating aEIF neurons without adaptation and with either a subthreshold or a spike-triggered adaptation current, respectively, and analyzed the network activity. The neurons were all either excitatory or inhibitory and weakly coupled.

Degree of network synchronization

To understand the biophysical relevance of the subthreshold and spike-triggered adaptation parameters,

Without adaptation,

A: Membrane potential

We further show how the PRC characteristics caused by the adaptation currents depend on the injected current

We show the effects of the adaptation currents

A–D: Stable (solid black) and unstable (dashed grey) phase locked states of coupled pairs of Traub neurons with identical PRCs, as a function of conductances

The effects of

In this work we studied the role of adaptation in the aEIF model as an endogenous neuronal mechanism that controls network dynamics. We described the effects of subthreshold and spike-triggered adaptation currents on the PRC in dependence of spike frequency. To provide insight into the synchronization tendencies of coupled neurons, we applied a common phase reduction technique and used the PRC to describe neuronal interaction

Conductance

PRCs determine synchronization properties of coupled oscillating neurons. When the synapses are fast compared to the oscillation period, the stability of the in-phase and anti-phase locked states (which always exist for pairs of identical neurons) can be “read off” the PRC for any mutual conduction delay, as we have demonstrated. A similar stability criterion that depends on the slopes of the PRCs at the phases at which the inputs are received has recently been derived for pairs of pulse-coupled oscillators

We have shown that, as long as synaptic delays are negligible and synaptic strengths equal, excitatory pairs synchronize if their PRCs are type II, as caused by

The activity of larger aEIF networks, simulated numerically, is consistent with the predictions of the behavior of pairs. In fact, knowledge on phase locking of coupled pairs helps to explain the observed network states. Both adaptation mediated PRC characteristics, i.e. a negative lobe or a pronounced right skew, favor synchronization in networks of excitatory neurons, in agreement with previous findings

Spike frequency has been shown to affect the skewness of PRCs, using type I integrate-and-fire neurons with adaptation

The adaptation currents

Our analysis of phase locked states is based on the assumption that synaptic interactions are weak. Experimental work lending support to this assumption has been reviewed in

Supplementary

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