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Conceived and designed the experiments: CGF VB MZ. Performed the experiments: CGF. Analyzed the data: CGF VB MZ. Contributed reagents/materials/analysis tools: MZ. Wrote the paper: CGF VB MZ.

The authors have declared that no competing interests exist.

Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.

Synchronization of the firing of neurons in the brain is related to many cognitive functions, such as recognizing faces, discriminating odors, and coordinating movement. It is therefore important to understand what properties of neuronal networks promote synchrony of neural firing. One measure that is often used to determine the contribution of individual neurons to network synchrony is called the phase response curve (PRC). PRCs describe how the timing of neuronal firing changes depending on when input, such as a synaptic signal, is received by the neuron. A characteristic of PRCs that has previously not been well understood is that they change dramatically as the neuron's firing frequency is modulated. This effect carries potential significance, since cognitive functions are often associated with specific frequencies of network activity in the brain. We showed computationally that the frequency dependence of PRCs can be explained by the relative timing of ionic membrane currents with respect to the time between spike firings. Our simulations also showed that the frequency dependence of neuronal PRCs leads to frequency-dependent changes in network synchronization that can be different for different neuron types. These results further our understanding of how synchronization is generated in the brain to support various cognitive functions.

Neuronal synchronization is thought to underlie spatiotemporal pattern formation in the healthy

To demonstrate the universality of the frequency-dependent effects on the neuronal PRC, we consider a reduced model neuron described by the Morris-Lecar equations

Our results provide important insight into differential changes in the propensity for network synchronization induced by the external modulation of neuronal frequency. As neuronal firing frequency changes, the changes in network spatiotemporal patterns depend upon the response characteristics of the individual cells in the network.

We used the Morris-Lecar model

The Type I and Type II neuronal models share the parameter values

The cortical pyramidal model neuron we employed was motivated by recent computational and experimental findings, as reported in

with

Activation of the

with

with

where

The slow, low-threshold

(A) Frequency-current curve for Type I and Type II Morris-Lecar model neurons. Note that the Type I cell can fire at arbitrarily low frequencies, while the Type II cell exhibits a non-zero frequency threshold. (B,C) Frequency dependence of PRCs for Morris-Lecar model neurons with Type I and Type II response characteristics. When the PRC was computed at different neuronal firing frequencies (different curves), amplitudes of phase shifts were attenuated, and the Type II neuron showed asymmetric attenuation of the phase advance and phase delay regions. (D) Frequency-current curves for Type I (

For both neuronal models,

In all network simulations, the number of neurons is 200, and the synapses are exclusively excitatory. The network connectivity pattern is constructed using the Watts-Strogatz architecture for Small World Networks

Synaptic current is transmitted from neuron

We employ two different methods to modulate network firing frequency in our simulations. The first is to simply modulate the supra-threshold value of

In order to model more biologically relevant environmental inputs, we also run simulations of cortical pyramidal neuronal networks in which frequency is modulated by stochastic input. All neurons are given the same constant sub-threshold baseline current, plus square current pulses randomly delivered to each neuron at a specified frequency

We monitor phase-synchronization of neuronal firing in our simulations using the mean phase coherence (MPC) measure,

where

We quantify phase-zero synchronization of a network by calculating the bursting measure

where

We first investigate the underlying cellular basis of the differential frequency effects on Type I and Type II PRCs. We show that the relative activation levels and timing of hyperpolarizing, potassium currents in relation to depolarizing currents play a crucial role in shaping the phase response of a neuron. We then show that individual neuronal spiking frequency modulates network synchrony in significantly different ways for networks consisting of Type I or Type II cells. Specifically, synchrony in Type I networks is affected very little by frequency modulation near threshold, whereas in Type II networks, synchrony falls dramatically as frequency increases above firing threshold. This effect is due to the disparity in the frequency-modulated attenuation of the PRCs of the two cell types. We first show this effect in excitatory networks composed of Morris-Lecar model neurons, and then investigate it in depth for excitatory networks consisting of model cortical pyramidal cells under acetylcholine modulation.

In both models, increasing frequency by increasing the constant applied current results in an attenutation of phase responses (

In both models, phase delays exist in the Type II parameter regimes because there is a voltage interval in which activation of an outward, hyperpolarizing current is greater than activation of the inward, depolarizing current. In the Type II Morris-Lecar model, the steady state activation curve of the

As firing frequency increases, the cycle trajectory passes through this

We further illustrate this point by modulating the speed of the gating variable controlling the delay-inducing potassium current in each model.

(A–C) Effects of modifying the speed of the potassium current in the Type II Morris-Lecar neuron, with increasing values of

A similar dependence of phase delay amplitude on the rate of the gating variable

In the cortical pyramidal neuron model, the amplitude of phase delays also depended on the rate of the

These results imply that appropriate selection of the rate of variables gating the intracellular currents mentioned above permits the recovery of specified PRC delay depths for different levels of external current.

(A) PRC profiles of the Type II Morris-Lecar neuron for three different values of

We analyzed network activity patterns in large-scale (N = 200) excitatory networks composed of Morris-Lecar model neurons with Type I and Type II PRCs under different network connectivity regimes. As described in the

Measures of network activity for simulations of large-scale (N = 200) excitatory networks of Morris-Lecar model neurons driven with various constant applied currents (different curves) for Type I (A,C,E) and Type II (B,D,F) cells. The synaptic coupling was set to

We first investigated synchronization properties of networks driven with constant applied currents, as in the Morris-Lecar network simulations. Every cell was driven with a constant current,

(A–F) Measures of network activity for simulations of large-scale (N = 200) excitatory networks of cortical pyramidal model neurons driven with varying constant applied currents for Type I (A,B,C) and Type II (D,E,F) cells. Synaptic weight was fixed at

We observed sharp differences between responses of the Type I and Type II networks to frequency modulation. As shown in

To show that these results were robust to network structure and coupling strength,

(A,B) Phase-zero synchrony (as measured by the bursting parameter,

In the high-coupling regime, where differences in steady-state synchronization between high- and low-frequency Type II networks were diminished, we investigated whether frequency might still affect the

(A) Bursting parameter

Finally,

(A,B) Differences in MPC between high- and low-frequency networks as a function of network re-wiring and synaptic weight for (A) Type I and (B) Type II networks composed of cortical pyramidal cells. Values of

After we demonstrated the distinct synchronization response properties of Type I and Type II networks stimulated by varying levels of constant current, we next investigated the more biologically relevant context of stochastic stimulation. Here random current pulses were used to simulate neuronal drive coming from other brain modalities.

(A,B) Average network frequency as a function of the re-wiring parameter for various values of

(A,B) Differences in bursting parameter

Phase locking again largely followed the same trend as bursting, with the difference in MPC assuming values near zero for most of parameter space in Type I networks (

We have shown that excitatory networks composed of neurons with either Type I or Type II PRC properties respond very differently to frequency modulation near firing threshold, with Type I network synchrony remaining largely unaffected by frequency modulation and Type II networks synchronizing much better at lower frequencies. This result is robust in virtually all network parameter regimes in which the network is capable of attaining any appreciable level of synchronization. While both Type I and Type II PRCs are modified by changes in frequency, only Type II PRCs change in qualitative profile. Specifically, the phase delay region, which is known to be critical in promoting synchrony, is severely attenuated. Increased frequency therefore tends to have little effect upon Type I networks, since there is no change in the PRC's contribution to synchrony, while in Type II networks it leads to depressed synchrony via the diminished phase delay region of the PRC. It should be noted that our simulations agreed with a large body of previous work showing that neurons with Type II membrane dynamics (as defined by the frequency-current curve) tend to synchronize better than neurons with Type I membrane dynamics, when coupled with excitation. Previous theoretical work indicates that when excitatory networks are driven with constant current, those composed of Type I neurons will not synchronize as well as those composed of Type II neurons

In this study, we focused on the implications for network synchronization of the observed frequency-dependence of PRCs. Our results suggest that the severe attenuation of the phase-delay region of Type II PRCs at increased firing frequencies contributes to the observed decline in network synchronization at such frequencies. Frequency-dependent modification of PRCs has been investigated before in complex, multi-compartment neuronal models

The frequency-dependent synchronization which we have described in this paper could potentially be involved in any cognitive process, functional or pathological, which involves spatiotemporal pattern formation of neuronal populations. For example, cholinergically-induced switching between sensitivity and insensitivity to frequency modulation could be important in proper memory consolidation during slow wave and REM sleep, two states that are characterized by differing levels of acetylcholine in cortical and hippocampal regions. Frequency-mediated synchrony could also play a part in the binding of signals from multiple sensory modalities. Gamma oscillations (20–80 Hz) in cortical networks are believed to be generated by synchronous activity of fast-spiking interneurons

At the same time, the importance of our results is not confined to these examples alone. Our findings point to the possibility that Type I and Type II excitatory networks function in two separate coding regimes, with Type I networks functioning in the rate coding regime and Type II networks functioning in the temporal coding regime, effectively acting as low-pass filters. Further experimental investigation into the interplay between cellular properties, frequency, and network synchronization is clearly required.