Spiking resonances in models with the same slow resonant and fast amplifying currents but different subthreshold dynamic properties

Low-dimensional models of single neurons: a review

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as subthreshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type.

Network resonance can be generated independently at distinct levels of neuronal organization

Resonance is defined as maximal response of a system to periodic inputs in a limited frequency band. Resonance may serve to optimize inter-neuronal communication, and has been observed at multiple levels of neuronal organization. However, it is unknown how neuronal resonance observed at the network level is generated and how network resonance depends on the properties of the network building blocks. Here, we first develop a metric for quantifying spike timing resonance in the presence of background noise, extending the notion of spiking resonance for in vivo experiments. Using conductance-based models, we find that network resonance can be inherited from resonances at other levels of organization, or be intrinsically generated by combining mechanisms across distinct levels. Resonance of membrane potential fluctuations, postsynaptic potentials, and single neuron spiking can each be generated independently of resonance at any other level and be propagated to the network level. At all levels of organization, interactions between processes that give rise to low- and high-pass filters generate the observed resonance. Intrinsic network resonance can be generated by the combination of filters belonging to different levels of organization. Inhibition-induced network resonance can emerge by inheritance from resonance of membrane potential fluctuations, and be sharpened by presynaptic high-pass filtering. Our results demonstrate a multiplicity of qualitatively different mechanisms that can generate resonance in neuronal systems, and provide analysis tools and a conceptual framework for the mechanistic investigation of network resonance in terms of circuit components, across levels of neuronal organization.

How one part of the brain responds to periodic input from another part depends on resonant circuit properties. Resonance is a basic property of physical systems, and has been experimentally observed at various levels of neuronal organization both in vitro and in vivo. Yet how resonance is generated in neuronal networks is largely unknown. In particular, whether resonance can be generated directly at the level of a network of spiking neurons remains to be determined. Using detailed biophysical modeling, we develop a conceptual framework according to which resonance at a given level of organization is generated by the interplay of low- and high-pass filters, implemented at either the same or across levels of neuronal organization. We tease apart representative, biophysically-plausible generative mechanisms of resonance at four different levels of organization: membrane potential fluctuations, single neuron spiking, synaptic transmission, and neuronal networks. We identify conditions under which resonance at one level can be inherited to another level of organization, provide conclusive evidence that resonance at each level can be generated without resonance at any other level, and describe a number of representative routes to network resonance. The proposed framework facilitates the investigation of resonance in neuronal systems.

The voltage and spiking responses of subthreshold resonant neurons to structured and fluctuating inputs: persistence and loss of resonance and variability

We systematically investigate the response of neurons to oscillatory currents and synaptic-like inputs and we extend our investigation to non-structured synaptic-like spiking inputs with more realistic distributions of presynaptic spike times. We use two types of chirp-like inputs consisting of (i) a sequence of cycles with discretely increasing frequencies over time, and (ii) a sequence having the same cycles arranged in an arbitrary order. We develop and use a number of frequency-dependent voltage response metrics to capture the different aspects of the voltage response, including the standard impedance (Z) and the peak-to-trough amplitude envelope ([Formula: see text]) profiles. We show that Z-resonant cells (cells that exhibit subthreshold resonance in response to sinusoidal inputs) also show [Formula: see text]-resonance in response to sinusoidal inputs, but generally do not (or do it very mildly) in response to square-wave and synaptic-like inputs. In the latter cases the resonant response using Z is not predictive of the preferred frequencies at which the neurons spike when the input amplitude is increased above subthreshold levels. We also show that responses to conductance-based synaptic-like inputs are attenuated as compared to the response to current-based synaptic-like inputs, thus providing an explanation to previous experimental results. These response patterns were strongly dependent on the intrinsic properties of the participating neurons, in particular whether the unperturbed Z-resonant cells had a stable node or a focus. In addition, we show that variability emerges in response to chirp-like inputs with arbitrarily ordered patterns where all signals (trials) in a given protocol have the same frequency content and the only source of uncertainty is the subset of all possible permutations of cycles chosen for a given protocol. This variability is the result of the multiple different ways in which the autonomous transient dynamics is activated across cycles in each signal (different cycle orderings) and across trials. We extend our results to include high-rate Poisson distributed current- and conductance-based synaptic inputs and compare them with similar results using additive Gaussian white noise. We show that the responses to both Poisson-distributed synaptic inputs are attenuated with respect to the responses to Gaussian white noise. For cells that exhibit oscillatory responses to Gaussian white noise (band-pass filters), the response to conductance-based synaptic inputs are low-pass filters, while the response to current-based synaptic inputs may remain band-pass filters, consistent with experimental findings. Our results shed light on the mechanisms of communication of oscillatory activity among neurons in a network via subthreshold oscillations and resonance and the generation of network resonance.

From mechanisms to functions: The role of theta and gamma coherence in the intrahippocampal circuits

Brain rhythms are essential for information processing in neuronal networks. Oscillations recorded in different brain regions can be synchronized and have a constant phase difference, that is, they can be coherent. Coherence between local field potential (LFP) signals from different brain regions may be correlated with the performance of cognitive tasks, indicating that these regions of the brain are jointly involved in the information processing. Why does coherence occur and how is it related to the information transfer between different regions of the hippocampal formation? In this article, we discuss possible mechanisms of theta and gamma coherence and its role in the hippocampus-dependent attention and memory processes, since theta and gamma rhythms are most pronounced in these processes. We review in vivo studies of interactions between different regions of the hippocampal formation in theta and gamma frequency bands. The key propositions of the review are as follows: (1) coherence emerges from synchronous postsynaptic currents in principal neurons as a result of synchronization of neuronal spike activity; (2) the synchronization of neuronal spike patterns in two regions of the hippocampal formation can be realized through induction or resonance; (3) coherence at a specific time point reflects the transfer of information between the regions of the hippocampal formation; (4) the physiological roles of theta and gamma coherence are different due to their different functions and mechanisms of generation. All hippocampal neurons are involved in theta activity, and theta coherence arranges the firing order of principal neurons throughout the hippocampal formation. In contrast, gamma coherence reflects the coupling of active neuronal ensembles. Overall, the coherence of LFPs between different areas of the brain is an important physiological process based on the synchronized neuronal firing, and it is essential for cooperative information processing.

Diversity amongst human cortical pyramidal neurons revealed via their sag currents and frequency preferences

In the human neocortex coherent interlaminar theta oscillations are driven by deep cortical layers, suggesting neurons in these layers exhibit distinct electrophysiological properties. To characterize this potential distinctiveness, we use in vitro whole-cell recordings from cortical layers 2 and 3 (L2&3), layer 3c (L3c) and layer 5 (L5) of the human cortex. Across all layers we observe notable heterogeneity, indicating human cortical pyramidal neurons are an electrophysiologically diverse population. L5 pyramidal cells are the most excitable of these neurons and exhibit the most prominent sag current (abolished by blockade of the hyperpolarization activated cation current, I_h). While subthreshold resonance is more common in L3c and L5, we rarely observe this resonance at frequencies greater than 2 Hz. However, the frequency dependent gain of L5 neurons reveals they are most adept at tracking both delta and theta frequency inputs, a unique feature that may indirectly be important for the generation of cortical theta oscillations.

Differential contributions of synaptic and intrinsic inhibitory currents to speech segmentation via flexible phase-locking in neural oscillators

Current hypotheses suggest that speech segmentation—the initial division and grouping of the speech stream into candidate phrases, syllables, and phonemes for further linguistic processing—is executed by a hierarchy of oscillators in auditory cortex. Theta (∼3-12 Hz) rhythms play a key role by phase-locking to recurring acoustic features marking syllable boundaries. Reliable synchronization to quasi-rhythmic inputs, whose variable frequency can dip below cortical theta frequencies (down to ∼1 Hz), requires “flexible” theta oscillators whose underlying neuronal mechanisms remain unknown. Using biophysical computational models, we found that the flexibility of phase-locking in neural oscillators depended on the types of hyperpolarizing currents that paced them. Simulated cortical theta oscillators flexibly phase-locked to slow inputs when these inputs caused both (i) spiking and (ii) the subsequent buildup of outward current sufficient to delay further spiking until the next input. The greatest flexibility in phase-locking arose from a synergistic interaction between intrinsic currents that was not replicated by synaptic currents at similar timescales. Flexibility in phase-locking enabled improved entrainment to speech input, optimal at mid-vocalic channels, which in turn supported syllabic-timescale segmentation through identification of vocalic nuclei. Our results suggest that synaptic and intrinsic inhibition contribute to frequency-restricted and -flexible phase-locking in neural oscillators, respectively. Their differential deployment may enable neural oscillators to play diverse roles, from reliable internal clocking to adaptive segmentation of quasi-regular sensory inputs like speech.

Oscillatory activity in auditory cortex is believed to play an important role in auditory and speech processing. One suggested function of these rhythms is to divide the speech stream into candidate phonemes, syllables, words, and phrases, to be matched with learned linguistic templates. This requires brain rhythms to flexibly synchronize with regular acoustic features of the speech stream. How neuronal circuits implement this task remains unknown. In this study, we explored the contribution of inhibitory currents to flexible phase-locking in neuronal theta oscillators, believed to perform initial syllabic segmentation. We found that a combination of specific intrinsic inhibitory currents at multiple timescales, present in a large class of cortical neurons, enabled exceptionally flexible phase-locking, which could be used to precisely segment speech by identifying vowels at mid-syllable. This suggests that the cells exhibiting these currents are a key component in the brain’s auditory and speech processing architecture.

Modeling Reveals Human-Rodent Differences in H-Current Kinetics Influencing Resonance in Cortical Layer 5 Neurons

While our understanding of human neurons is often inferred from rodent data, inter-species differences between neurons can be captured by building cellular models specifically from human data. This includes understanding differences at the level of ion channels and their implications for human brain function. Thus, we here present a full spiking, biophysically detailed multi-compartment model of a human layer 5 (L5) cortical pyramidal cell. Model development was primarily based on morphological and electrophysiological data from the same human L5 neuron, avoiding confounds of experimental variability. Focus was placed on describing the behavior of the hyperpolarization-activated cation (h-) channel, given increasing interest in this channel due to its role in pacemaking and differentiating cell types. We ensured that the model exhibited post-inhibitory rebound spiking considering its relationship with the h-current, along with other general spiking characteristics. The model was validated against data not used in its development, which highlighted distinctly slower kinetics of the human h-current relative to the rodent setting. We linked the lack of subthreshold resonance observed in human L5 neurons to these human-specific h-current kinetics. This work shows that it is possible and necessary to build human-specific biophysical neuron models in order to understand human brain dynamics.

Optimization of Efficient Neuron Models With Realistic Firing Dynamics. The Case of the Cerebellar Granule Cell

Biologically relevant large-scale computational models currently represent one of the main methods in neuroscience for studying information processing primitives of brain areas. However, biologically realistic neuron models tend to be computationally heavy and thus prevent these models from being part of brain-area models including thousands or even millions of neurons. The cerebellar input layer represents a canonical example of large scale networks. In particular, the cerebellar granule cells, the most numerous cells in the whole mammalian brain, have been proposed as playing a pivotal role in the creation of somato-sensorial information representations. Enhanced burst frequency (spiking resonance) in the granule cells has been proposed as facilitating the input signal transmission at the theta-frequency band (4–12 Hz), but the functional role of this cell feature in the operation of the granular layer remains largely unclear. This study aims to develop a methodological pipeline for creating neuron models that maintain biological realism and computational efficiency whilst capturing essential aspects of single-neuron processing. Therefore, we selected a light computational neuron model template (the adaptive-exponential integrate-and-fire model), whose parameters were progressively refined using an automatic parameter tuning with evolutionary algorithms (EAs). The resulting point-neuron models are suitable for reproducing the main firing properties of a realistic granule cell from electrophysiological measurements, including the spiking resonance at the theta-frequency band, repetitive firing according to a specified intensity-frequency (I-F) curve and delayed firing under current-pulse stimulation. Interestingly, the proposed model also reproduced some other emergent properties (namely, silent at rest, rheobase and negligible adaptation under depolarizing currents) even though these properties were not set in the EA as a target in the fitness function (FF), proving that these features are compatible even in computationally simple models. The proposed methodology represents a valuable tool for adjusting AdEx models according to a FF defined in the spiking regime and based on biological data. These models are appropriate for future research of the functional implication of bursting resonance at the theta band in large-scale granular layer network models.

Prefrontal oscillations modulate the propagation of neuronal activity required for working memory

Cognition involves using attended information, maintained in working memory (WM), to guide action. During a cognitive task, a correct response requires flexible, selective gating so that only the appropriate information flows from WM to downstream effectors that carry out the response. In this work, we used biophysically-detailed modeling to explore the hypothesis that network oscillations in prefrontal cortex (PFC), leveraging local inhibition, can independently gate responses to items in WM. The key role of local inhibition was to control the period between spike bursts in the outputs, and to produce an oscillatory response no matter whether the WM item was maintained in an asynchronous or oscillatory state. We found that the WM item that induced an oscillatory population response in the PFC output layer with the shortest period between spike bursts was most reliably propagated. The network resonant frequency (i.e., the input frequency that produces the largest response) of the output layer can be flexibly tuned by varying the excitability of deep layer principal cells. Our model suggests that experimentally-observed modulation of PFC beta-frequency (15-30 Hz) and gamma-frequency (30-80 Hz) oscillations could leverage network resonance and local inhibition to govern the flexible routing of signals in service to cognitive processes like gating outputs from working memory and the selection of rule-based actions. Importantly, we show for the first time that nonspecific changes in deep layer excitability can tune the output gate's resonant frequency, enabling the specific selection of signals encoded by populations in asynchronous or fast oscillatory states. More generally, this represents a dynamic mechanism by which adjusting network excitability can govern the propagation of asynchronous and oscillatory signals throughout neocortex.

Broadband Entrainment of Striatal Low-Threshold Spike Interneurons

Striatal interneurons and spiny projection (SP) neurons are differentially tuned to spectral components of their input. Previous studies showed that spike responses of somatostatin/NPY-expressing low threshold spike (LTS) interneurons have broad frequency tuning, setting these cells apart from other striatal GABAergic interneurons and SP neurons. We investigated the mechanism of LTS interneuron spiking resonance and its relationship to non-spiking membrane impedance resonance, finding that abolition of impedance resonance did not alter spiking resonance. Because LTS interneurons are pacemakers whose rhythmic firing is perturbed by synaptic input, we tested the hypothesis that their spiking resonance arises from their phase resetting properties. Phase resetting curves (PRCs) were measured in LTS interneurons and SP neurons and used to make phase-oscillator models of both cell types. The models reproduced the broad tuning of LTS interneurons, and the differences from SP neurons. The spectral components of the PRC predicted each cell's sensitivity to corresponding input frequencies. LTS interneuron PRCs contain larger high-frequency components than SP neuron PRCs, providing enhanced responses to input frequencies above the cells' average firing rates. Thus, LTS cells can be entrained by input oscillations to which SP neurons are less responsive. These findings suggest that feedforward inhibition by LTS interneurons may regulate SP neurons' entrainment by oscillatory afferents.

Asymmetrical voltage response in resonant neurons shaped by nonlinearities

The conventional impedance profile of a neuron can identify the presence of resonance and other properties of the neuronal response to oscillatory inputs, such as nonlinear response amplifications, but it cannot distinguish other nonlinear properties such as asymmetries in the shape of the voltage response envelope. Experimental observations have shown that the response of neurons to oscillatory inputs preferentially enhances either the upper or lower part of the voltage envelope in different frequency bands. These asymmetric voltage responses arise in a neuron model when it is submitted to high enough amplitude oscillatory currents of variable frequencies. We show how the nonlinearities associated to different ionic currents or present in the model as captured by its voltage equation lead to asymmetrical response and how high amplitude oscillatory currents emphasize this response. We propose a geometrical explanation for the phenomenon where asymmetries result not only from nonlinearities in their activation curves but also from nonlinearites captured by the nullclines in the phase-plane diagram and from the system's time-scale separation. In addition, we identify an unexpected frequency-dependent pattern which develops in the gating variables of these currents and is a product of strong nonlinearities in the system as we show by controlling such behavior by manipulating the activation curve parameters. The results reported in this paper shed light on the ionic mechanisms by which brain embedded neurons process oscillatory information.

Frequency-dependent responses of neuronal models to oscillatory inputs in current versus voltage clamp

Action potential generation in neurons depends on a membrane potential threshold and therefore on how subthreshold inputs influence this voltage. In oscillatory networks, for example, many neuron types have been shown to produce membrane potential ([Formula: see text]) resonance: a maximum subthreshold response to oscillatory inputs at a nonzero frequency. Resonance is usually measured by recording [Formula: see text] in response to a sinusoidal current ([Formula: see text]), applied at different frequencies (f), an experimental setting known as current clamp (I-clamp). Several recent studies, however, use the voltage clamp (V-clamp) method to control [Formula: see text] with a sinusoidal input at different frequencies [[Formula: see text]] and measure the total membrane current ([Formula: see text]). The two methods obey systems of differential equations of different dimensionality, and while I-clamp provides a measure of electrical impedance [[Formula: see text]], V-clamp measures admittance [[Formula: see text]]. We analyze the relationship between these two measurement techniques. We show that, despite different dimensionality, in linear systems the two measures are equivalent: [Formula: see text]. However, nonlinear model neurons produce different values for Z and [Formula: see text]. In particular, nonlinearities in the voltage equation produce a much larger difference between these two quantities than those in equations of recovery variables that describe activation and inactivation kinetics. Neurons are inherently nonlinear, and notably, with ionic currents that amplify resonance, the voltage clamp technique severely underestimates the current clamp response. We demonstrate this difference experimentally using the PD neurons in the crab stomatogastric ganglion. These findings are instructive for researchers who explore cellular mechanisms of neuronal oscillations.

Frequency-dependent entrainment of striatal fast-spiking interneurons

Striatal fast-spiking interneurons (FSIs) fire in variable-length runs of action potentials at 20-200 spikes/s separated by pauses. In vivo, or with fluctuating applied current, both runs and pauses become briefer and more variable. During runs, spikes are entrained specifically to gamma-frequency components of the input fluctuations. We stimulated parvalbumin-expressing striatal FSIs in mouse brain slices with broadband noise currents added to direct current steps and measured spike entrainment across all frequencies. As the constant current level was increased, FSIs produced longer runs and showed sharper frequency tuning, with best entrainment at the stimulus frequency matching their intrarun firing rate. We separated the contributions of previous spikes from that of the fluctuating stimulus, revealing a strong contribution of previous action potentials to gamma-frequency entrainment. In contrast, after subtraction of the effect inherited from the previous spike, the remaining stimulus contribution to spike generation was less sharply tuned, showing a larger contribution of lower frequencies. The frequency specificity of entrainment within a run was reproduced with a phase resetting model based on experimentally measured phase resetting curves of the same FSIs. In the model, broadly tuned phase entrainment for the first spike in a run evolved into sharply tuned gamma entrainment over the next few spikes. The data and modeling results indicate that for FSIs firing in brief runs and pauses firing within runs is entrained by gamma-frequency components of the input, whereas the onset timing of runs may be sensitive to a wider range of stimulus frequency components.NEW & NOTEWORTHY Specific types of neurons entrain their spikes to particular oscillation frequencies in their synaptic input. This entrainment is commonly understood in terms of the subthreshold voltage response, but how this translates to spiking is not clear. We show that in striatal fast-spiking interneurons, entrainment to gamma-frequency input depends on rhythmic spike runs and is explained by the phase resetting curve, whereas run initiation can be triggered by a broad range of input frequencies.

Active membrane conductances and morphology of a collision detection neuron broaden its impedance profile and improve discrimination of input synchrony

How neurons filter and integrate their complex patterns of synaptic inputs is central to their role in neural information processing. Synaptic filtering and integration are shaped by the frequency-dependent neuronal membrane impedance. Using single and dual dendritic recordings in vivo, pharmacology, and computational modeling, we characterized the membrane impedance of a collision detection neuron in the grasshopper Schistocerca americana. This neuron, the lobula giant movement detector (LGMD), exhibits consistent impedance properties across frequencies and membrane potentials. Two common active conductances g_H and g_M, mediated respectively by hyperpolarization-activated cyclic nucleotide-gated (HCN) channels and by muscarine-sensitive M-type K⁺ channels, promote broadband integration with high temporal precision over the LGMD's natural range of membrane potentials and synaptic input frequencies. Additionally, we found that a model based on the LGMD's branching morphology increased the gain and decreased the delay associated with the mapping of synaptic input currents to membrane potential. More generally, this was true for a wide range of model neuron morphologies, including those of neocortical pyramidal neurons and cerebellar Purkinje cells. These findings show the unexpected role played by two widespread active conductances and by dendritic morphology in shaping synaptic integration.NEW & NOTEWORTHY Neuronal filtering and integration of synaptic input patterns depend on the electrochemical properties of dendrites. We used an identified collision detection neuron in grasshoppers to examine how its morphology and two conductances affect its membrane impedance in relation to the computations it performs. The neuronal properties examined are ubiquitous and therefore promote a general understanding of neuronal computations, including those in the human brain.

Membrane potential resonance in non-oscillatory neurons interacts with synaptic connectivity to produce network oscillations

Several neuron types have been shown to exhibit (subthreshold) membrane potential resonance (MPR), defined as the occurrence of a peak in their voltage amplitude response to oscillatory input currents at a preferred (resonant) frequency. MPR has been investigated both experimentally and theoretically. However, whether MPR is simply an epiphenomenon or it plays a functional role for the generation of neuronal network oscillations and how the latent time scales present in individual, non-oscillatory cells affect the properties of the oscillatory networks in which they are embedded are open questions. We address these issues by investigating a minimal network model consisting of (i) a non-oscillatory linear resonator (band-pass filter) with 2D dynamics, (ii) a passive cell (low-pass filter) with 1D linear dynamics, and (iii) nonlinear graded synaptic connections (excitatory or inhibitory) with instantaneous dynamics. We demonstrate that (i) the network oscillations crucially depend on the presence of MPR in the resonator, (ii) they are amplified by the network connectivity, (iii) they develop relaxation oscillations for high enough levels of mutual inhibition/excitation, and (iv) the network frequency monotonically depends on the resonators resonant frequency. We explain these phenomena using a reduced adapted version of the classical phase-plane analysis that helps uncovering the type of effective network nonlinearities that contribute to the generation of network oscillations. We extend our results to networks having cells with 2D dynamics. Our results have direct implications for network models of firing rate type and other biological oscillatory networks (e.g, biochemical, genetic).

Oscillations and Spike Entrainment

Oscillatory input to networks, as indicated by field potentials, must entrain neuronal firing to be a causal agent in brain activity. Even when the oscillatory input is prominent, entrainment of firing is not a foregone conclusion but depends on the intrinsic dynamics of the postsynaptic neurons, including cell type-specific resonances, and background firing rates. Within any local network of neurons, only a subset of neurons may have their firing entrained by an oscillating synaptic input, and oscillations of different frequency may engage separate subsets of neurons.

M-Current Expands the Range of Gamma Frequency Inputs to Which a Neuronal Target Entrains

Theta (4-8 Hz) and gamma (30-80 Hz) rhythms in the brain are commonly associated with memory and learning (Kahana in J Neurosci 26:1669-1672, 2006; Quilichini et al. in J Neurosci 30:11128-11142, 2010). The precision of co-firing between neurons and incoming inputs is critical in these cognitive functions. We consider an inhibitory neuron model with M-current under forcing from gamma pulses and a sinusoidal current of theta frequency. The M-current has a long time constant (∼90 ms) and it has been shown to generate resonance at theta frequencies (Hutcheon and Yarom in Trends Neurosci 23:216-222, 2000; Hu et al. in J Physiol 545:783-805, 2002). We have found that this slow M-current contributes to the precise co-firing between the network and fast gamma pulses in the presence of a slow sinusoidal forcing. The M-current expands the phase-locking frequency range of the network, counteracts the slow theta forcing, and admits bistability in some parameter range. The effects of the M-current balancing the theta forcing are reduced if the sinusoidal current is faster than the theta frequency band. We characterize the dynamical mechanisms underlying the role of the M-current in enabling a network to be entrained to gamma frequency inputs using averaging methods, geometric singular perturbation theory, and bifurcation analysis.

Flexible resonance in prefrontal networks with strong feedback inhibition

Oscillations are ubiquitous features of brain dynamics that undergo task-related changes in synchrony, power, and frequency. The impact of those changes on target networks is poorly understood. In this work, we used a biophysically detailed model of prefrontal cortex (PFC) to explore the effects of varying the spike rate, synchrony, and waveform of strong oscillatory inputs on the behavior of cortical networks driven by them. Interacting populations of excitatory and inhibitory neurons with strong feedback inhibition are inhibition-based network oscillators that exhibit resonance (i.e., larger responses to preferred input frequencies). We quantified network responses in terms of mean firing rates and the population frequency of network oscillation; and characterized their behavior in terms of the natural response to asynchronous input and the resonant response to oscillatory inputs. We show that strong feedback inhibition causes the PFC to generate internal (natural) oscillations in the beta/gamma frequency range (>15 Hz) and to maximize principal cell spiking in response to external oscillations at slightly higher frequencies. Importantly, we found that the fastest oscillation frequency that can be relayed by the network maximizes local inhibition and is equal to a frequency even higher than that which maximizes the firing rate of excitatory cells; we call this phenomenon population frequency resonance. This form of resonance is shown to determine the optimal driving frequency for suppressing responses to asynchronous activity. Lastly, we demonstrate that the natural and resonant frequencies can be tuned by changes in neuronal excitability, the duration of feedback inhibition, and dynamic properties of the input. Our results predict that PFC networks are tuned for generating and selectively responding to beta- and gamma-rhythmic signals due to the natural and resonant properties of inhibition-based oscillators. They also suggest strategies for optimizing transcranial stimulation and using oscillatory networks in neuromorphic engineering.