Resonant Interneurons Can Increase Robustness of Gamma Oscillations

Firing rate models for gamma oscillations in I-I and E-I networks

Firing rate models for describing the mean-field activities of neuronal ensembles can be used effectively to study network function and dynamics, including synchronization and rhythmicity of excitatory-inhibitory populations. However, traditional Wilson-Cowan-like models, even when extended to include an explicit dynamic synaptic activation variable, are found unable to capture some dynamics such as Interneuronal Network Gamma oscillations (ING). Use of an explicit delay is helpful in simulations at the expense of complicating mathematical analysis. We resolve this issue by introducing a dynamic variable, u, that acts as an effective delay in the negative feedback loop between firing rate (r) and synaptic gating of inhibition (s). In effect, u endows synaptic activation with second order dynamics. With linear stability analysis, numerical branch-tracking and simulations, we show that our r-u-s rate model captures some key qualitative features of spiking network models for ING. We also propose an alternative formulation, a v-u-s model, in which mean membrane potential v satisfies an averaged current-balance equation. Furthermore, we extend the framework to E-I networks. With our six-variable v-u-s model, we demonstrate in firing rate models the transition from Pyramidal-Interneuronal Network Gamma (PING) to ING by increasing the external drive to the inhibitory population without adjusting synaptic weights. Having PING and ING available in a single network, without invoking synaptic blockers, is plausible and natural for explaining the emergence and transition of two different types of gamma oscillations.

Persistent Interruption in Parvalbumin-Positive Inhibitory Interneurons: Biophysical and Mathematical Mechanisms

Persistent activity in excitatory pyramidal cells (PYRs) is a putative mechanism for maintaining memory traces during working memory. We have recently demonstrated persistent interruption of firing in fast-spiking parvalbumin-expressing interneurons (PV-INs), a phenomenon that could serve as a substrate for persistent activity in PYRs through disinhibition lasting hundreds of milliseconds. Here, we find that hippocampal CA1 PV-INs exhibit type 2 excitability, like striatal and neocortical PV-INs. Modeling and mathematical analysis showed that the slowly inactivating potassium current KV1 contributes to type 2 excitability, enables the multiple firing regimes observed experimentally in PV-INs, and provides a mechanism for robust persistent interruption of firing. Using a fast/slow separation of times scales approach with the KV1 inactivation variable as a bifurcation parameter shows that the initial inhibitory stimulus stops repetitive firing by moving the membrane potential trajectory onto a coexisting stable fixed point corresponding to a nonspiking quiescent state. As KV1 inactivation decays, the trajectory follows the branch of stable fixed points until it crosses a subcritical Hopf bifurcation (HB) and then spirals out into repetitive firing. In a model describing entorhinal cortical PV-INs without KV1, interruption of firing could be achieved by taking advantage of the bistability inherent in type 2 excitability based on a subcritical HB, but the interruption was not robust to noise. Persistent interruption of firing is therefore broadly applicable to PV-INs in different brain regions but is only made robust to noise in the presence of a slow variable, KV1 inactivation.

Persistent Interruption in Parvalbumin Positive Inhibitory Interneurons: Biophysical and Mathematical Mechanisms

Persistent activity in principal cells is a putative mechanism for maintaining memory traces during working memory. We recently demonstrated persistent interruption of firing in fast-spiking parvalbumin-expressing interneurons (PV-INs), a phenomenon which could serve as a substrate for persistent activity in principal cells through disinhibition lasting hundreds of milliseconds. Here, we find that hippocampal CA1 PV-INs exhibit type 2 excitability, like striatal and neocortical PV-INs. Modelling and mathematical analysis showed that the slowly inactivating potassium current Kv1 contributes to type 2 excitability, enables the multiple firing regimes observed experimentally in PV-INs, and provides a mechanism for robust persistent interruption of firing. Using a fast/slow separation of times scales approach with the Kv1 inactivation variable as a bifurcation parameter shows that the initial inhibitory stimulus stops repetitive firing by moving the membrane potential trajectory onto a co-existing stable fixed point corresponding to a non-spiking quiescent state. As Kv1 inactivation decays, the trajectory follows the branch of stable fixed points until it crosses a subcritical Hopf bifurcation then spirals out into repetitive firing. In a model describing entorhinal cortical PV-INs without Kv1, interruption of firing could be achieved by taking advantage of the bistability inherent in type 2 excitability based on a subcritical Hopf bifurcation, but the interruption was not robust to noise. Persistent interruption of firing is therefore broadly applicable to PV-INs in different brain regions but is only made robust to noise in the presence of a slow variable.

Rodents' visual gamma as a biomarker of pathological neural conditions

Neural gamma oscillations (indicatively 30-100 Hz) are ubiquitous: they are associated with a broad range of functions in multiple cortical areas and across many animal species. Experimental and computational works established gamma rhythms as a global emergent property of neuronal networks generated by the balanced and coordinated interaction of excitation and inhibition. Coherently, gamma activity is strongly influenced by the alterations of synaptic dynamics which are often associated with pathological neural dysfunctions. We argue therefore that these oscillations are an optimal biomarker for probing the mechanism of cortical dysfunctions. Gamma oscillations are also highly sensitive to external stimuli in sensory cortices, especially the primary visual cortex (V1), where the stimulus dependence of gamma oscillations has been thoroughly investigated. Gamma manipulation by visual stimuli tuning is particularly easy in rodents, which have become a standard animal model for investigating the effects of network alterations on gamma oscillations. Overall, gamma in the rodents' visual cortex offers an accessible probe on dysfunctional information processing in pathological conditions. Beyond vision-related dysfunctions, alterations of gamma oscillations in rodents were indeed also reported in neural deficits such as migraine, epilepsy and neurodegenerative or neuropsychiatric conditions such as Alzheimer's, schizophrenia and autism spectrum disorders. Altogether, the connections between visual cortical gamma activity and physio-pathological conditions in rodent models underscore the potential of gamma oscillations as markers of neuronal (dys)functioning.

Synchrony in Networks of Type 2 Interneurons Is More Robust to Noise with Hyperpolarizing Inhibition Compared to Shunting Inhibition in Both the Stochastic Population Oscillator and the Coupled Oscillator Regimes

Synchronization in the gamma band (25-150 Hz) is mediated by PV+ inhibitory interneurons, and evidence is accumulating for the essential role of gamma oscillations in cognition. Oscillations can arise in inhibitory networks via synaptic interactions between individual oscillatory neurons (mean-driven) or via strong recurrent inhibition that destabilizes the stationary background firing rate in the fluctuation-driven balanced state, causing an oscillation in the population firing rate. Previous theoretical work focused on model neurons with Hodgkin's Type 1 excitability (integrators) connected by current-based synapses. Here we show that networks comprised of simple Type 2 oscillators (resonators) exhibit a supercritical Hopf bifurcation between synchrony and asynchrony and a gradual transition via cycle skipping from coupled oscillators to stochastic population oscillator (SPO), as previously shown for Type 1. We extended our analysis to homogeneous networks with conductance rather than current based synapses and found that networks with hyperpolarizing inhibitory synapses were more robust to noise than those with shunting synapses, both in the coupled oscillator and SPO regime. Assuming that reversal potentials are uniformly distributed between shunting and hyperpolarized values, as observed in one experimental study, converting synapses to purely hyperpolarizing favored synchrony in all cases, whereas conversion to purely shunting synapses made synchrony less robust except at very high conductance strengths. In mature neurons the synaptic reversal potential is controlled by chloride cotransporters that control the intracellular concentrations of chloride and bicarbonate ions, suggesting these transporters as a potential therapeutic target to enhance gamma synchrony and cognition.

Physiological features of parvalbumin-expressing GABAergic interneurons contributing to high-frequency oscillations in the cerebral cortex

Parvalbumin-expressing (PV+) inhibitory interneurons drive gamma oscillations (30-80 Hz), which underlie higher cognitive functions. In this review, we discuss two groups/aspects of fundamental properties of PV+ interneurons. In the first group (dubbed Before Axon), we list properties representing optimal synaptic integration in PV+ interneurons designed to support fast oscillations. For example: [i] Information can neither enter nor leave the neocortex without the engagement of fast PV+ -mediated inhibition; [ii] Voltage responses in PV+ interneuron dendrites integrate linearly to reduce impact of the fluctuations in the afferent drive; and [iii] Reversed somatodendritic Rm gradient accelerates the time courses of synaptic potentials arriving at the soma. In the second group (dubbed After Axon), we list morphological and biophysical properties responsible for (a) short synaptic delays, and (b) efficient postsynaptic outcomes. For example: [i] Fast-spiking ability that allows PV+ interneurons to outpace other cortical neurons (pyramidal neurons). [ii] Myelinated axon (which is only found in the PV+ subclass of interneurons) to secure fast-spiking at the initial axon segment; and [iii] Inhibitory autapses - autoinhibition, which assures brief biphasic voltage transients and supports postinhibitory rebounds. Recent advent of scientific tools, such as viral strategies to target PV cells and the ability to monitor PV cells via in vivo imaging during behavior, will aid in defining the role of PV cells in the CNS. Given the link between PV+ interneurons and cognition, in the future, it would be useful to carry out physiological recordings in the PV+ cell type selectively and characterize if and how psychiatric and neurological diseases affect initiation and propagation of electrical signals in this cortical sub-circuit. Voltage imaging may allow fast recordings of electrical signals from many PV+ interneurons simultaneously.

Synchrony in Networks of Type 2 Interneurons is More Robust to Noise with Hyperpolarizing Inhibition Compared to Shunting Inhibition in Both the Stochastic Population Oscillator and the Coupled Oscillator Regimes

Synchronization in the gamma band (30-80 Hz) is mediated by PV+ inhibitory interneurons, and evidence is accumulating for the essential role of gamma oscillations in cognition. Oscillations can arise in inhibitory networks via synaptic interactions between individual oscillatory neurons (mean-driven) or via strong recurrent inhibition that destabilizes the stationary background firing rate in the fluctuation-driven balanced state, causing an oscillation in the population firing rate. Previous theoretical work focused on model neurons with Hodgkin's type 1 excitability (integrators) connected by current-based synapses. Here we show that networks comprised of simple type 2 oscillators (resonators) exhibit a supercritical Hopf bifurcation between synchrony and asynchrony and a gradual transition via cycle skipping from coupled oscillators to stochastic population oscillator, as previously shown for type 1. We extended our analysis to homogeneous networks with conductance rather than current based synapses and found that networks with hyperpolarizing inhibitory synapses were more robust to noise than those with shunting synapses, both in the coupled oscillator and stochastic population oscillator regime. Assuming that reversal potentials are uniformly distributed between shunting and hyperpolarized values, as observed in one experimental study, converting synapses to purely hyperpolarizing favored synchrony in all cases, whereas conversion to purely shunting synapses made synchrony less robust except at very high conductance strengths. In mature neurons the synaptic reversal potential is controlled by chloride cotransporters that control the intracellular concentrations of chloride and bicarbonate ions, suggesting these transporters as a potential therapeutic target to enhance gamma synchrony and cognition.

Significance Statement

Brain rhythms in the gamma frequency band (30-80 Hz) depend on the activity of inhibitory interneurons and evidence for a causal role for gamma oscillations in cognitive functions is accumulating. Here we extend previous studies on synchronization mechanisms to interneurons that have an abrupt threshold frequency below which they cannot sustain firing. In addition to current based synapses, we examined inhibitory networks with conductance based synapses. We found that if the reversal potential for inhibition was below the average membrane potential (hyperpolarizing), synchrony was more robust to noise than if the reversal potential was very close to the average potential (shunting). These results have implications for therapies to ameliorate cognitive deficits.

Influence of inhibitory autapses on synchronization of inhibitory network gamma oscillations

A recent experimental study showed that inhibitory autapses favor firing synchronization of parvalbumin interneurons in the neocortex during gamma oscillations. In the present paper, to provide a comprehensive and deep understanding to the experimental observation, the influence of inhibitory autapses on synchronization of interneuronal network gamma oscillations is theoretically investigated. Weak, middle, and strong synchronizations of a globally inhibitory coupled network composed of Wang-Buzsáki model without autapses appear at the bottom-left, middle, and top-right of the parameter plane with the conductance (gsyn) and the decay constant (τsyn) of inhibitory synapses taken as the x-axis and y-axis, respectively. After introducing inhibitory autapses, the border between the strong and middle synchronizations in the (gsyn, τsyn) plane moves to the top-right with increasing the conductance (gaut) and the decay constant (τaut) of autapses, due to that interspike interval of the single neuron becomes longer, leading to that larger τsyn is needed to ensure the strong synchronization. Then, the synchronization degree of middle and strong synchronizations around the border in the (gsyn, τsyn) plane decreases, while of strong synchronization in the remaining region remains unchanged. The synchronization degree of weak synchronization increases with increasing τaut and gaut, due to that the inhibitory autaptic current becomes strong and long to facilitate synchronization. The enhancement of weak synchronization modulated by inhibitory autapses is also simulated in the random, small-world, and scale-free networks, which may provide explanations to the experimental observation. These results present complex dynamics of synchronization modulated by inhibitory autapses, which needs future experimental demonstrations.

Propagation of Oscillations in the Indirect Pathway of the Basal Ganglia

Oscillatory signals propagate in the basal ganglia from prototypic neurons in the external globus pallidus (GPe) to their target neurons in the substantia nigra pars reticulata (SNr), internal pallidal segment, and subthalamic nucleus. Neurons in the GPe fire spontaneously, so oscillatory input signals can be encoded as changes in timing of action potentials within an ongoing spike train. When GPe neurons were driven by an oscillatory current in male and female mice, these spike-timing changes produced spike-oscillation coherence over a range of frequencies extending at least to 100 Hz. Using the known kinetics of the GPe→SNr synapse, we calculated the postsynaptic currents that would be generated in SNr neurons from the recorded GPe spike trains. The ongoing synaptic barrage from spontaneous firing, frequency-dependent short-term depression, and stochastic fluctuations at the synapse embed the input oscillation into a noisy sequence of synaptic currents in the SNr. The oscillatory component of the resulting synaptic current must compete with the noisy spontaneous synaptic barrage for control of postsynaptic SNr neurons, which have their own frequency-dependent sensitivities. Despite this, SNr neurons subjected to synaptic conductance changes generated from recorded GPe neuron firing patterns also became coherent with oscillations over a broad range of frequencies. The presynaptic, synaptic, and postsynaptic frequency sensitivities were all dependent on the firing rates of presynaptic and postsynaptic neurons. Firing rate changes, often assumed to be the propagating signal in these circuits, do not encode most oscillation frequencies, but instead determine which signal frequencies propagate effectively and which are suppressed.SIGNIFICANCE STATEMENT Oscillations are present in all the basal ganglia nuclei, include a range of frequencies, and change over the course of learning and behavior. Exaggerated oscillations are a hallmark of basal ganglia pathologies, and each has a specific frequency range. Because of its position as a hub in the basal ganglia circuitry, the globus pallidus is a candidate origin for oscillations propagating between nuclei. We imposed low-amplitude oscillations on individual globus pallidus neurons at specific frequencies and measured the coherence between the oscillation and firing as a function of frequency. We then used these responses to measure the effectiveness of oscillatory propagation to other basal ganglia nuclei. Propagation was effective for oscillation frequencies as high as 100 Hz.

Nonlinear mechanism for the enhanced bursting activities induced by fast inhibitory autapse and reduced activities by fast excitatory autapse

The paradoxical phenomena that excitatory modulation does not enhance but reduces or inhibitory modulation not suppresses but promotes neural firing activities have attracted increasing attention. In the present study, paradoxical phenomena induced by both fast excitatory and inhibitory autapses in a "Fold/Big Homoclinic" bursting are simulated, and the corresponding nonlinear and biophysical mechanisms are presented. Firstly, the enhanced conductance of excitatory autapse induces the number of spikes per burst and firing rate reduced, while the enhanced inhibitory autapse cause both indicators increased. Secondly, with fast-slow variable dissection, the burst of bursting is identified to locate between a fold bifurcation and a big saddle-homoclinic orbit bifurcation of the fast subsystem. Enhanced excitatory or inhibitory autapses cannot induce changes of both bifurcation points, i.e., burst width. However, width of slow variable between two successive spikes within a burst becomes wider for the excitatory autapse and narrower for the inhibitory autapse, resulting in the less and more spikes per burst, respectively. Last, the autaptic current of fast autapse mainly plays a role during the peak of action potential, differing from the slow autaptic current with exponential decay, which can play roles following the peak of action potential. The fast excitatory autaptic current enhances the amplitude of the action potential and reduces the repolarization of the action potential to lengthen the interspike interval (ISI) of the spiking of the fast subsystem, resulting in the wide width of slow variable between successive spikes. The fast inhibitory autaptic current reduces the amplitude of action potential and ISI of spiking, resulting in narrow width of slow variable. The novel example of the paradoxical responses for both fast modulations and nonlinear mechanism extend the contents of neurodynamics, which presents potential functions of the fast autapse.

Timing to be precise? An overview of spike timing-dependent plasticity, brain rhythmicity, and glial cells interplay within neuronal circuits

In the mammalian brain information processing and storage rely on the complex coding and decoding events performed by neuronal networks. These actions are based on the computational ability of neurons and their functional engagement in neuronal assemblies where precise timing of action potential firing is crucial. Neuronal circuits manage a myriad of spatially and temporally overlapping inputs to compute specific outputs that are proposed to underly memory traces formation, sensory perception, and cognitive behaviors. Spike-timing-dependent plasticity (STDP) and electrical brain rhythms are suggested to underlie such functions while the physiological evidence of assembly structures and mechanisms driving both processes continues to be scarce. Here, we review foundational and current evidence on timing precision and cooperative neuronal electrical activity driving STDP and brain rhythms, their interactions, and the emerging role of glial cells in such processes. We also provide an overview of their cognitive correlates and discuss current limitations and controversies, future perspectives on experimental approaches, and their application in humans.

Over and above frequency: Gamma oscillations as units of neural circuit operations

Gamma oscillations (∼30-150 Hz) are widespread correlates of neural circuit functions. These network activity patterns have been described across multiple animal species, brain structures, and behaviors, and are usually identified based on their spectral peak frequency. Yet, despite intensive investigation, whether gamma oscillations implement causal mechanisms of specific brain functions or represent a general dynamic mode of neural circuit operation remains unclear. In this perspective, we review recent advances in the study of gamma oscillations toward a deeper understanding of their cellular mechanisms, neural pathways, and functional roles. We discuss that a given gamma rhythm does not per se implement any specific cognitive function but rather constitutes an activity motif reporting the cellular substrates, communication channels, and computational operations underlying information processing in its generating brain circuit. Accordingly, we propose shifting the attention from a frequency-based to a circuit-level definition of gamma oscillations.

Theta-gamma phase amplitude coupling in a hippocampal CA1 microcircuit

Phase amplitude coupling (PAC) between slow and fast oscillations is found throughout the brain and plays important functional roles. Its neural origin remains unclear. Experimental findings are often puzzling and sometimes contradictory. Most computational models rely on pairs of pacemaker neurons or neural populations tuned at different frequencies to produce PAC. Here, using a data-driven model of a hippocampal microcircuit, we demonstrate that PAC can naturally emerge from a single feedback mechanism involving an inhibitory and excitatory neuron population, which interplay to generate theta frequency periodic bursts of higher frequency gamma. The model suggests the conditions under which a CA1 microcircuit can operate to elicit theta-gamma PAC, and highlights the modulatory role of OLM and PVBC cells, recurrent connectivity, and short term synaptic plasticity. Surprisingly, the results suggest the experimentally testable prediction that the generation of the slow population oscillation requires the fast one and cannot occur without it.

Interneuronal network model of theta-nested fast oscillations predicts differential effects of heterogeneity, gap junctions and short term depression for hyperpolarizing versus shunting inhibition

Theta and gamma oscillations in the hippocampus have been hypothesized to play a role in the encoding and retrieval of memories. Recently, it was shown that an intrinsic fast gamma mechanism in medial entorhinal cortex can be recruited by optogenetic stimulation at theta frequencies, which can persist with fast excitatory synaptic transmission blocked, suggesting a contribution of interneuronal network gamma (ING). We calibrated the passive and active properties of a 100-neuron model network to capture the range of passive properties and frequency/current relationships of experimentally recorded PV+ neurons in the medial entorhinal cortex (mEC). The strength and probabilities of chemical and electrical synapses were also calibrated using paired recordings, as were the kinetics and short-term depression (STD) of the chemical synapses. Gap junctions that contribute a noticeable fraction of the input resistance were required for synchrony with hyperpolarizing inhibition; these networks exhibited theta-nested high frequency oscillations similar to the putative ING observed experimentally in the optogenetically-driven PV-ChR2 mice. With STD included in the model, the network desynchronized at frequencies above ~200 Hz, so for sufficiently strong drive, fast oscillations were only observed before the peak of the theta. Because hyperpolarizing synapses provide a synchronizing drive that contributes to robustness in the presence of heterogeneity, synchronization decreases as the hyperpolarizing inhibition becomes weaker. In contrast, networks with shunting inhibition required non-physiological levels of gap junctions to synchronize using conduction delays within the measured range.

Dynamical properties of neuromorphic Josephson junctions

Neuromorphic computing exploits the dynamical analogy between many physical systems and neuron biophysics. Superconductor systems, in particular, are excellent candidates for neuromorphic devices due to their capacity to operate at great speeds and with low energy dissipation compared to their silicon counterparts. In this paper, we revisit a prior work on Josephson Junction-based neurons to identify the exact dynamical mechanisms underlying the system's neuronlike properties and reveal complex behaviors which are relevant for neurocomputation and the design of superconducting neuromorphic devices. Our paper lies at the intersection of superconducting physics and theoretical neuroscience, both viewed under a common framework-that of nonlinear dynamics theory.

Parvalbumin interneuron dendrites enhance gamma oscillations

Dendrites are essential determinants of the input-output relationship of single neurons, but their role in network computations is not well understood. Here, we use a combination of dendritic patch-clamp recordings and in silico modeling to determine how dendrites of parvalbumin (PV)-expressing basket cells contribute to network oscillations in the gamma frequency band. Simultaneous soma-dendrite recordings from PV basket cells in the dentate gyrus reveal that the slope, or gain, of the dendritic input-output relationship is exceptionally low, thereby reducing the cell's sensitivity to changes in its input. By simulating gamma oscillations in detailed network models, we demonstrate that the low gain is key to increase spike synchrony in PV basket cell assemblies when cells are driven by spatially and temporally heterogeneous synaptic inputs. These results highlight the role of inhibitory neuron dendrites in synchronized network oscillations.

Low EEG Gamma Entropy and Glucose Hypometabolism After Corpus Callosotomy Predicts Seizure Outcome After Subsequent Surgery

Background

Patients with generalized epilepsy who had lateralized EEG abnormalities after corpus callosotomy (CC) occasionally undergo subsequent surgeries to control intractable epilepsy.

Objectives

This study evaluated retrospectively the combination of EEG multiscale entropy (MSE) and FDG-PET for identifying lateralization of the epileptogenic zone after CC.

Methods

This study included 14 patients with pharmacoresistant epilepsy who underwent curative epilepsy surgery after CC. Interictal scalp EEG and FDG-PET obtained after CC were investigated to determine (1) whether the MSE calculated from the EEG and FDG-PET findings was lateralized to the surgical side, and (2) whether the lateralization was associated with seizure outcomes.

Results

Seizure reduction rate was higher in patients with lateralized findings to the surgical side than those without (MSE: p < 0.05, FDG-PET: p < 0.05, both: p < 0.01). Seizure free rate was higher in patients with lateralized findings in both MSE and FDG-PET than in those without (p < 0.05).

Conclusions

This study demonstrated that patients with lateralization of MSE and FDG-PET to the surgical side had better seizure outcomes. The combination of MSE and conventional FDG-PET may help to select surgical candidates for additional surgery after CC with good postoperative seizure outcomes.

Kinetics and Connectivity Properties of Parvalbumin- and Somatostatin-Positive Inhibition in Layer 2/3 Medial Entorhinal Cortex

Parvalbumin-positive (Pvalb⁺) and somatostatin-positive (Sst⁺) cells are the two largest subgroups of inhibitory interneurons. Studies in visual cortex indicate that synaptic connections between Pvalb⁺ cells are common while connections between Sst⁺ interneurons have not been observed. The inhibitory connectivity and kinetics of these two interneuron subpopulations, however, have not been characterized in medial entorhinal cortex (mEC). Using fluorescence-guided paired recordings in mouse brain slices from interneurons and excitatory cells in layer 2/3 mEC, we found that, unlike neocortical measures, Sst⁺ cells inhibit each other, albeit with a lower probability than Pvalb⁺ cells (18% vs 36% for unidirectional connections). Gap junction connections were also more frequent between Pvalb⁺ cells than between Sst⁺ cells. Pvalb⁺ cells inhibited each other with larger conductances, smaller decay time constants, and shorter delays. Similarly, synaptic connections between Pvalb⁺ and excitatory cells were more likely and expressed faster decay times and shorter delays than those between Sst⁺ and excitatory cells. Inhibitory cells exhibited smaller synaptic decay time constants between interneurons than on their excitatory targets. Inhibition between interneurons also depressed faster, and to a greater extent. Finally, inhibition onto layer 2 pyramidal and stellate cells originating from Pvalb⁺ interneurons were very similar, with no significant differences in connection likelihood, inhibitory amplitude, and decay time. A model of short-term depression fitted to the data indicates that recovery time constants for refilling the available pool are in the range of 50-150 ms and that the fraction of the available pool released on each spike is in the range 0.2-0.5.

Bri2 BRICHOS chaperone rescues impaired fast-spiking interneuron behavior and neuronal network dynamics in an AD mouse model in vitro

Synchronized and properly balanced electrical activity of neurons is the basis for the brain's ability to process information, to learn, and to remember. In Alzheimer's disease (AD), which causes cognitive decline in patients, this synchronization and balance is disturbed by the accumulation of neuropathological biomarkers such as amyloid-beta peptide (Aβ42). Failure of Aβ42 clearance mechanisms as well as desynchronization of crucial neuronal classes such as fast-spiking interneurons (FSN) are root causes for the disruption of the cognition-relevant gamma brain rhythm (30-80 Hz) and consequent cognitive impairment observed in AD. Here we show that recombinant BRICHOS molecular chaperone domains from ProSP-C or Bri2, which interfere with Aβ42 aggregation, can rescue the gamma rhythm. We demonstrate that Aβ42 progressively decreases gamma oscillation power and rhythmicity, disrupts the inhibition/excitation balance in pyramidal cells, and desynchronizes FSN firing during gamma oscillations in the hippocampal CA3 network of mice. Application of the more efficacious Bri2 BRICHOS chaperone rescued the cellular and neuronal network performance from all ongoing Aβ42-induced functional impairments. Collectively, our findings offer critical missing data to explain the importance of FSN for normal network function and underscore the therapeutic potential of Bri2 BRICHOS to rescue the disruption of cognition-relevant brain rhythms in AD.

Resonating neurons stabilize heterogeneous grid-cell networks

A central theme that governs the functional design of biological networks is their ability to sustain stable function despite widespread parametric variability. Here, we investigated the impact of distinct forms of biological heterogeneities on the stability of a two-dimensional continuous attractor network (CAN) implicated in grid-patterned activity generation. We show that increasing degrees of biological heterogeneities progressively disrupted the emergence of grid-patterned activity and resulted in progressively large perturbations in low-frequency neural activity. We postulated that targeted suppression of low-frequency perturbations could ameliorate heterogeneity-induced disruptions of grid-patterned activity. To test this, we introduced intrinsic resonance, a physiological mechanism to suppress low-frequency activity, either by adding an additional high-pass filter (phenomenological) or by incorporating a slow negative feedback loop (mechanistic) into our model neurons. Strikingly, CAN models with resonating neurons were resilient to the incorporation of heterogeneities and exhibited stable grid-patterned firing. We found CAN models with mechanistic resonators to be more effective in targeted suppression of low-frequency activity, with the slow kinetics of the negative feedback loop essential in stabilizing these networks. As low-frequency perturbations (1/f noise) are pervasive across biological systems, our analyses suggest a universal role for mechanisms that suppress low-frequency activity in stabilizing heterogeneous biological networks.

Ablation of p75NTR signaling strengthens gamma-theta rhythm interaction and counteracts Aβ-induced degradation of neuronal dynamics in mouse hippocampus in vitro

Gamma and theta brain rhythms play important roles in cognition and their interaction can affect gamma oscillation features. Hippocampal theta oscillations depend on cholinergic and GABAergic input from the medial septum-diagonal band of Broca. These projecting neurons undergo degeneration during aging and maintain high levels of neurotrophin receptor p75 (p75^NTR). p75^NTR mediates both apoptosis and survival and its expression is increased in Alzheimer's disease (AD) patients. Here, we investigate the importance of p75^NTR for the cholinergic input to the hippocampus. Performing extracellular recordings in brain slices from p75^NTR knockout mice (p75^-/-) in presence of the muscarinic agonist carbachol, we find that gamma oscillation power and rhythmicity are increased compared to wild-type (WT) mice. Furthermore, gamma activity is more phase-locked to the underlying theta rhythm, which renders a stronger coupling of both rhythms. On the cellular level, we find that fast-spiking interneurons (FSNs) fire more synchronized to a preferred gamma phase in p75^-/- mice. The excitatory input onto FSN is more rhythmic displaying a higher similarity with the concomitant gamma rhythm. Notably, the ablation of p75^NTR counteracts the Aβ-induced degradation of gamma oscillations and its nesting within the underlying theta rhythm. Our results show that the lack of p75^NTR signaling could promote stronger cholinergic modulation of the hippocampal gamma rhythm, suggesting an involvement of p75^NTR in the downregulation of cognition-relevant hippocampal network dynamics in pathologies. Moreover, functional data provided here suggest p75^NTR as a suitable target in the search for efficacious treatments to counteract the loss of cognitive function observed in amyloid-driven pathologies such as AD.

Excitability and Threshold Mechanism for Enhanced Neuronal Response Induced by Inhibition Preceding Excitation

Postinhibitory facilitation (PIF) of neural firing presents a paradoxical phenomenon that the inhibitory effect induces enhancement instead of reduction of the firing activity, which plays important roles in sound location of the auditory nervous system, awaited theoretical explanations. In the present paper, excitability and threshold mechanism for the PIF phenomenon is presented in the Morris-Lecar model with type I, II, and III excitabilities. Firstly, compared with the purely excitatory stimulations applied to the steady state, the inhibitory preceding excitatory stimulation to form pairs induces the firing rate increased for type II and III excitabilities instead of type I excitability, when the interval between the inhibitory and excitatory stimulation within each pair is suitable. Secondly, the threshold mechanism for the PIF phenomenon is acquired. For type II and III excitabilities, the inhibitory stimulation induces subthreshold oscillations around the steady state. During the middle and ending phase of the ascending part and the beginning phase of the descending part within a period of the subthreshold oscillations, the threshold to evoke an action potential by an excitatory stimulation becomes weaker, which is the cause for the PIF phenomenon. Last, a theoretical estimation for the range of the interval between the inhibitory and excitatory stimulation for the PIF phenomenon is acquired, which approximates half of the intrinsic period of the subthreshold oscillations for the relatively strong stimulations and becomes narrower for the relatively weak stimulations. The interval for the PIF phenomenon is much shorter for type III excitability, which is closer to the experiment observation, due to the shorter period of the subthreshold oscillations. The results present the excitability and threshold mechanism for the PIF phenomenon, which provide comprehensive and deep explanations to the PIF phenomenon.

Information processing and energy efficiency of temperature-sensitive Morris-Lecar neuron

In biological organisms, the temperature is an important factor, which affects the motion of micro-particles and biochemical reaction. In current work, we investigate the effect of temperature on the capacity of information processing and the energy efficiency of the Hodgkin's three basic classes of neurons. Increasing the temperature, both of the total entropy and information rate would maintain nearly as constant, and then decrease rapidly and fall to zero. Moreover, energy consumption is reduced as the temperature increases. However, the energy consumption of the class 1 neuron is remarkably greater than that of class 2 and 3 neurons with the same temperature. It is also interesting that the class 3 neuron consumes less energy than that of class 1 and 2 neurons, but generates the same value of total entropy and information rate for the same condition.

Shunting Inhibition Improves Synchronization in Heterogeneous Inhibitory Interneuronal Networks with Type 1 Excitability Whereas Hyperpolarizing Inhibition Is Better for Type 2 Excitability

All-to-all homogeneous networks of inhibitory neurons synchronize completely under the right conditions; however, many modeling studies have shown that biological levels of heterogeneity disrupt synchrony. Our fundamental scientific question is “how can neurons maintain partial synchrony in the presence of heterogeneity and noise?” A particular subset of strongly interconnected interneurons, the PV+ fast-spiking (FS) basket neurons, are strongly implicated in γ oscillations and in phase locking of nested γ oscillations to theta. Their excitability type apparently varies between brain regions: in CA1 and the dentate gyrus they have type 1 excitability, meaning that they can fire arbitrarily slowly, whereas in the striatum and cortex they have type 2 excitability, meaning that there is a frequency thresh old below which they cannot sustain repetitive firing. We constrained the models to study the effect of excitability type (more precisely bifurcation type) in isolation from all other factors. We use sparsely connected, heterogeneous, noisy networks with synaptic delays to show that synchronization properties, namely the resistance to suppression and the strength of theta phase to γ amplitude coupling, are strongly dependent on the pairing of excitability type with the type of inhibition. Shunting inhibition performs better for type 1 and hyperpolarizing inhibition for type 2. γ Oscillations and their nesting within theta oscillations are thought to subserve cognitive functions like memory encoding and recall; therefore, it is important to understand the contribution of intrinsic properties to these rhythms.

Heterogeneities in intrinsic excitability and frequency-dependent response properties of granule cells across the blades of the rat dentate gyrus

The dentate gyrus (DG), the input gate to the hippocampus proper, is anatomically segregated into three different sectors, namely, the suprapyramidal blade, the crest region, and the infrapyramidal blade. Although there are well-established differences between these sectors in terms of neuronal morphology, connectivity patterns, and activity levels, differences in electrophysiological properties of granule cells within these sectors have remained unexplored. Here, employing somatic whole cell patch-clamp recordings from the rat DG, we demonstrate that granule cells in these sectors manifest considerable heterogeneities in their intrinsic excitability, temporal summation, action potential characteristics, and frequency-dependent response properties. Across sectors, these neurons showed positive temporal summation of their responses to inputs mimicking excitatory postsynaptic currents and showed little to no sag in their voltage responses to pulse currents. Consistently, the impedance amplitude profile manifested low-pass characteristics and the impedance phase profile lacked positive phase values at all measured frequencies and voltages and for all sectors. Granule cells in all sectors exhibited class I excitability, with broadly linear firing rate profiles, and granule cells in the crest region fired significantly fewer action potentials compared with those in the infrapyramidal blade. Finally, we found weak pairwise correlations across the 18 different measurements obtained individually from each of the three sectors, providing evidence that these measurements are indeed reporting distinct aspects of neuronal physiology. Together, our analyses show that granule cells act as integrators of afferent information and emphasize the need to account for the considerable physiological heterogeneities in assessing their roles in information encoding and processing.NEW & NOTEWORTHY We employed whole cell patch-clamp recordings from granule cells in the three subregions of the rat dentate gyrus to demonstrate considerable heterogeneities in their intrinsic excitability, temporal summation, action potential characteristics, and frequency-dependent response properties. Across sectors, granule cells did not express membrane potential resonance, and their impedance profiles lacked inductive phase leads at all measured frequencies. Our analyses also show that granule cells manifest class I excitability characteristics, categorizing them as integrators of afferent information.

Inhibitory Network Bistability Explains Increased Interneuronal Activity Prior to Seizure Onset

Recent experimental literature has revealed that GABAergic interneurons exhibit increased activity prior to seizure onset, alongside additional evidence that such activity is synchronous and may arise abruptly. These findings have led some to hypothesize that this interneuronal activity may serve a causal role in driving the sudden change in brain activity that heralds seizure onset. However, the mechanisms predisposing an inhibitory network toward increased activity, specifically prior to ictogenesis, without a permanent change to inputs to the system remain unknown. We address this question by comparing simulated inhibitory networks containing control interneurons and networks containing hyperexcitable interneurons modeled to mimic treatment with 4-Aminopyridine (4-AP), an agent commonly used to model seizures in vivo and in vitro. Our in silico study demonstrates that model inhibitory networks with 4-AP interneurons are more prone than their control counterparts to exist in a bistable state in which asynchronously firing networks can abruptly transition into synchrony driven by a brief perturbation. This transition into synchrony brings about a corresponding increase in overall firing rate. We further show that perturbations driving this transition could arise in vivo from background excitatory synaptic activity in the cortex. Thus, we propose that bistability explains the increase in interneuron activity observed experimentally prior to seizure via a transition from incoherent to coherent dynamics. Moreover, bistability explains why inhibitory networks containing hyperexcitable interneurons are more vulnerable to this change in dynamics, and how such networks can undergo a transition without a permanent change in the drive. We note that while our comparisons are between networks of control and ictogenic neurons, the conclusions drawn specifically relate to the unusual dynamics that arise prior to seizure, and not seizure onset itself. However, providing a mechanistic explanation for this phenomenon specifically in a pro-ictogenic setting generates experimentally testable hypotheses regarding the role of inhibitory neurons in pre-ictal neural dynamics, and motivates further computational research into mechanisms underlying a newly hypothesized multi-step pathway to seizure initiated by inhibition.

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.

Human cerebrospinal fluid promotes spontaneous gamma oscillations in the hippocampus in vitro

Gamma oscillations (30-80 Hz) are fast network activity patterns frequently linked to cognition. They are commonly studied in hippocampal brain slices in vitro, where they can be evoked via pharmacological activation of various receptor families. One limitation of this approach is that neuronal activity is studied in a highly artificial extracellular fluid environment, as provided by artificial cerebrospinal fluid (aCSF). Here, we examine the influence of human cerebrospinal fluid (hCSF) on kainate-evoked and spontaneous gamma oscillations in mouse hippocampus. We show that hCSF, as compared to aCSF of matched electrolyte and glucose composition, increases the power of kainate-evoked gamma oscillations and induces spontaneous gamma activity in areas CA3 and CA1 that is reversed by washout. Bath application of atropine entirely abolished hCSF-induced gamma oscillations, indicating critical contribution from muscarinic acetylcholine receptor-mediated signaling. In separate whole-cell patch clamp recordings from rat hippocampus, hCSF increased theta resonance frequency and strength in pyramidal cells along with enhancement of h-current (I_h ) amplitude. We found no evidence of intrinsic gamma frequency resonance at baseline (aCSF) among fast-spiking interneurons, and this was not altered by hCSF. However, hCSF increased the excitability of fast-spiking interneurons, which likely contributed to gamma rhythmogenesis. Our findings show that hCSF promotes network gamma oscillations in the hippocampus in vitro and suggest that neuromodulators distributed in CSF could have significant influence on neuronal network activity in vivo.

Firing rate models for gamma oscillations

Gamma oscillations are readily observed in a variety of brain regions during both waking and sleeping states. Computational models of gamma oscillations typically involve simulations of large networks of synaptically coupled spiking units. These networks can exhibit strongly synchronized gamma behavior, whereby neurons fire in near synchrony on every cycle, or weakly modulated gamma behavior, corresponding to stochastic, sparse firing of the individual units on each cycle of the population gamma rhythm. These spiking models offer valuable biophysical descriptions of gamma oscillations; however, because they involve many individual neuronal units they are limited in their ability to communicate general network-level dynamics. Here we demonstrate that few-variable firing rate models with established synaptic timescales can account for both strongly synchronized and weakly modulated gamma oscillations. These models go beyond the classical formulations of rate models by including at least two dynamic variables per population: firing rate and synaptic activation. The models' flexibility to capture the broad range of gamma behavior depends directly on the timescales that represent recruitment of the excitatory and inhibitory firing rates. In particular, we find that weakly modulated gamma oscillations occur robustly when the recruitment timescale of inhibition is faster than that of excitation. We present our findings by using an extended Wilson-Cowan model and a rate model derived from a network of quadratic integrate-and-fire neurons. These biophysical rate models capture the range of weakly modulated and coherent gamma oscillations observed in spiking network models, while additionally allowing for greater tractability and systems analysis. NEW & NOTEWORTHY Here we develop simple and tractable models of gamma oscillations, a dynamic feature observed throughout much of the brain with significant correlates to behavior and cognitive performance in a variety of experimental contexts. Our models depend on only a few dynamic variables per population, but despite this they qualitatively capture features observed in previous biophysical models of gamma oscillations that involve many individual spiking units.

A fingerprint of the epileptogenic zone in human epilepsies

Defining a bio-electrical marker for the brain area responsible for initiating a seizure remains an unsolved problem. Fast gamma activity has been identified as the most specific marker for seizure onset, but conflicting results have been reported. In this study, we describe an alternative marker, based on an objective description of interictal to ictal transition, with the aim of identifying a time-frequency pattern or ‘fingerprint’ that can differentiate the epileptogenic zone from areas of propagation. Seventeen patients who underwent stereoelectroencephalography were included in the study. Each had seizure onset characterized by sustained gamma activity and were seizure-free after tailored resection or laser ablation. We postulated that the epileptogenic zone was always located inside the resection region based on seizure freedom following surgery. To characterize the ictal frequency pattern, we applied the Morlet wavelet transform to data from each pair of adjacent intracerebral electrode contacts. Based on a visual assessment of the time-frequency plots, we hypothesized that a specific time-frequency pattern in the epileptogenic zone should include a combination of (i) sharp transients or spikes; preceding (ii) multiband fast activity concurrent; with (iii) suppression of lower frequencies. To test this hypothesis, we developed software that automatically extracted each of these features from the time-frequency data. We then used a support vector machine to classify each contact-pair as being within epileptogenic zone or not, based on these features. Our machine learning system identified this pattern in 15 of 17 patients. The total number of identified contacts across all patients was 64, with 58 localized inside the resected area. Subsequent quantitative analysis showed strong correlation between maximum frequency of fast activity and suppression inside the resection but not outside. We did not observe significant discrimination power using only the maximum frequency or the timing of fast activity to differentiate contacts either between resected and non-resected regions or between contacts identified as epileptogenic versus non-epileptogenic. Instead of identifying a single frequency or a single timing trait, we observed the more complex pattern described above that distinguishes the epileptogenic zone. This pattern encompasses interictal to ictal transition and may extend until seizure end. Its time-frequency characteristics can be explained in light of recent models emphasizing the role of fast inhibitory interneurons acting on pyramidal cells as a prominent mechanism in seizure triggering. The pattern clearly differentiates the epileptogenic zone from areas of propagation and, as such, represents an epileptogenic zone ‘fingerprint’.

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).

Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity

We show how to predict whether a neural network will exhibit global synchrony (a one-cluster state) or a two-cluster state based on the assumption of pulsatile coupling and critically dependent upon the phase response curve (PRC) generated by the appropriate perturbation from a partner cluster. Our results hold for a monotonically increasing (meaning longer delays as the phase increases) PRC, which likely characterizes inhibitory fast-spiking basket and cortical low-threshold-spiking interneurons in response to strong inhibition. Conduction delays stabilize synchrony for this PRC shape, whereas they destroy two-cluster states, the former by avoiding a destabilizing discontinuity and the latter by approaching it. With conduction delays, stronger coupling strength can promote a one-cluster state, so the weak coupling limit is not applicable here. We show how jitter can destabilize global synchrony but not a two-cluster state. Local stability of global synchrony in an all-to-all network does not guarantee that global synchrony can be observed in an appropriately scaled sparsely connected network; the basin of attraction can be inferred from the PRC and must be sufficiently large. Two-cluster synchrony is not obviously different from one-cluster synchrony in the presence of noise and may be the actual substrate for oscillations observed in the local field potential (LFP) and the electroencephalogram (EEG) in situations where global synchrony is not possible. Transitions between cluster states may change the frequency of the rhythms observed in the LFP or EEG. Transitions between cluster states within an inhibitory subnetwork may allow more effective recruitment of pyramidal neurons into the network rhythm. NEW & NOTEWORTHY We show that jitter induced by sparse connectivity can destabilize global synchrony but not a two-cluster state with two smaller clusters firing alternately. On the other hand, conduction delays stabilize synchrony and destroy two-cluster states. These results hold if each cluster exhibits a phase response curve similar to one that characterizes fast-spiking basket and cortical low-threshold-spiking cells for strong inhibition. Either a two-cluster or a one-cluster state might provide the oscillatory substrate for neural computations.

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.

Degeneracy in the robust expression of spectral selectivity, subthreshold oscillations, and intrinsic excitability of entorhinal stellate cells

Biological heterogeneities are ubiquitous and play critical roles in the emergence of physiology at multiple scales. Although neurons in layer II (LII) of the medial entorhinal cortex (MEC) express heterogeneities in channel properties, the impact of such heterogeneities on the robustness of their cellular-scale physiology has not been assessed. Here, we performed a 55-parameter stochastic search spanning nine voltage- or calcium-activated channels to assess the impact of channel heterogeneities on the concomitant emergence of 10 in vitro electrophysiological characteristics of LII stellate cells (SCs). We generated 150,000 models and found a heterogeneous subpopulation of 449 valid models to robustly match all electrophysiological signatures. We employed this heterogeneous population to demonstrate the emergence of cellular-scale degeneracy in SCs, whereby disparate parametric combinations expressing weak pairwise correlations resulted in similar models. We then assessed the impact of virtually knocking out each channel from all valid models and demonstrate that the mapping between channels and measurements was many-to-many, a critical requirement for the expression of degeneracy. Finally, we quantitatively predict that the spike-triggered average of SCs should be endowed with theta-frequency spectral selectivity and coincidence detection capabilities in the fast gamma-band. We postulate this fast gamma-band coincidence detection as an instance of cellular-scale-efficient coding, whereby SC response characteristics match the dominant oscillatory signals in LII MEC. The heterogeneous population of valid SC models built here unveils the robust emergence of cellular-scale physiology despite significant channel heterogeneities, and forms an efficacious substrate for evaluating the impact of biological heterogeneities on entorhinal network function.

NEW & NOTEWORTHY We assessed the impact of heterogeneities in channel properties on the robustness of cellular-scale physiology of medial entorhinal cortical stellate neurons. We demonstrate that neuronal models with disparate channel combinations were endowed with similar physiological characteristics, as a consequence of the many-to-many mapping between channel properties and the physiological characteristics that they modulate. We predict that the spike-triggered average of stellate cells should be endowed with theta-frequency spectral selectivity and fast gamma-band coincidence detection capabilities.

Dynamical Mechanism of Hyperpolarization-Activated Non-specific Cation Current Induced Resonance and Spike-Timing Precision in a Neuronal Model

Hyperpolarization-activated cyclic nucleotide-gated cation current (I_h) plays important roles in the achievement of many physiological/pathological functions in the nervous system by modulating the electrophysiological activities, such as the rebound (spike) to hyperpolarization stimulations, subthreshold membrane resonance to sinusoidal currents, and spike-timing precision to stochastic factors. In the present paper, with increasing g_h (conductance of I_h), the rebound (spike) and subthreshold resonance appear and become stronger, and the variability of the interspike intervals (ISIs) becomes lower, i.e., the enhancement of spike-timing precision, which are simulated in a conductance-based theoretical model and well explained by the nonlinear concept of bifurcation. With increasing g_h, the stable node to stable focus, to coexistence behavior, and to firing via the codimension-1 bifurcations (Hopf bifurcation, saddle-node bifurcation, saddle-node bifurcations on an invariant circle, and saddle homoclinic orbit) and codimension-2 bifurcations such as Bogdanov-Takens (BT) point related to the transition between saddle-node and Hopf bifurcations, are acquired with 1- and 2-parameter bifurcation analysis. The decrease of variability of ISIs with increasing g_h is induced by the fast decrease of the standard deviation of ISIs, which is related to the increase of the capacity of resisting noisy disturbance due to the firing becomes far away from the bifurcation point. The enhancement of the rebound (spike) with increasing g_h builds up a relationship to the decrease of the capacity of resisting disturbance like the hyperpolarization stimulus as the resting state approaches the bifurcation point. The “typical”-resonance and non-resonance appear in the parameter region of the stable focus and node far away from the bifurcation points, respectively. The complex or “strange” dynamics, such as the “weak”-resonance for the stable node near the transition point between the stable node and focus and the non-resonance for the stable focus close to the codimension-1 and −2 bifurcation points, are discussed.

Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.

Population models describing the average activity of large neuronal ensembles are a powerful mathematical tool to investigate the principles underlying cooperative function of large neuronal systems. However, these models do not properly describe the phenomenon of spike synchrony in networks of neurons. In particular, they fail to capture the onset of synchronous oscillations in networks of inhibitory neurons. We show that this limitation is due to a voltage-dependent synchronization mechanism which is naturally present in spiking neuron models but not captured by traditional firing rate equations. Here we investigate a novel set of macroscopic equations which incorporate both firing rate and membrane potential dynamics, and that correctly generate fast inhibition-based synchronous oscillations. In the limit of slow-synaptic processing oscillations are suppressed, and the model reduces to an equation formally equivalent to the Wilson-Cowan model.

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

The generation of spiking resonances in neurons (preferred spiking responses to oscillatory inputs) requires the interplay of the intrinsic ionic currents that operate at the subthreshold voltage level and the spiking mechanisms. Combinations of the same types of ionic currents in different parameter regimes may give rise to different types of nonlinearities in the voltage equation (e.g., parabolic- and cubic-like), generating subthreshold (membrane potential) oscillations patterns with different properties. These nonlinearities are not apparent in the model equations, but can be uncovered by plotting the voltage nullclines in the phase-plane diagram. We investigate the spiking resonant properties of conductance-based models that are biophysically equivalent at the subthreshold level (same ionic currents), but dynamically different (parabolic- and cubic-like voltage nullclines). As a case study we consider a model having a persistent sodium and a hyperpolarization-activated (h-) currents, which exhibits subthreshold resonance in the theta frequency band. We unfold the concept of spiking resonance into evoked and output spiking resonance. The former focuses on the input frequencies that are able to generate spikes, while the latter focuses on the output spiking frequencies regardless of the input frequency that generated these spikes. A cell can exhibit one or both types of resonances. We also measure spiking phasonance, which is an extension of subthreshold phasonance (zero-phase-shift response to oscillatory inputs) to the spiking regime. The subthreshold resonant properties of both types of models are communicated to the spiking regime for low enough input amplitudes as the voltage response for the subthreshold resonant frequency band raises above threshold. For higher input amplitudes evoked spiking resonance is no longer present in these models, but output spiking resonance is present primarily in the parabolic-like model due to a cycle skipping mechanism (involving mixed-mode oscillations), while the cubic-like model shows a better 1:1 entrainment. We use dynamical systems tools to explain the underlying mechanisms and the mechanistic differences between the resonance types. Our results demonstrate that the effective time scales that operate at the subthreshold regime to generate intrinsic subthreshold oscillations, mixed-mode oscillations and subthreshold resonance do not necessarily determine the existence of a preferred spiking response to oscillatory inputs in the same frequency band. The results discussed in this paper highlight both the complexity of the suprathreshold responses to oscillatory inputs in neurons having resonant and amplifying currents with different time scales and the fact that the identity of the participating ionic currents is not enough to predict the resulting patterns, but additional dynamic information, captured by the geometric properties of the phase-space diagram, is needed.

Unsupervised learning of temporal features for word categorization in a spiking neural network model of the auditory brain

The nature of the code used in the auditory cortex to represent complex auditory stimuli, such as naturally spoken words, remains a matter of debate. Here we argue that such representations are encoded by stable spatio-temporal patterns of firing within cell assemblies known as polychronous groups, or PGs. We develop a physiologically grounded, unsupervised spiking neural network model of the auditory brain with local, biologically realistic, spike-time dependent plasticity (STDP) learning, and show that the plastic cortical layers of the network develop PGs which convey substantially more information about the speaker independent identity of two naturally spoken word stimuli than does rate encoding that ignores the precise spike timings. We furthermore demonstrate that such informative PGs can only develop if the input spatio-temporal spike patterns to the plastic cortical areas of the model are relatively stable.

Transitions between classes of neuronal excitability and bifurcations induced by autapse

Neuronal excitabilities behave as the basic and important dynamics related to the transitions between firing and resting states, and are characterized by distinct bifurcation types and spiking frequency responses. Switches between class I and II excitabilities induced by modulations outside the neuron (for example, modulation to M-type potassium current) have been one of the most concerning issues in both electrophysiology and nonlinear dynamics. In the present paper, we identified switches between 2 classes of excitability and firing frequency responses when an autapse, which widely exists in real nervous systems and plays important roles via self-feedback, is introduced into the Morris-Lecar (ML) model neuron. The transition from class I to class II excitability and from class II to class I spiking frequency responses were respectively induced by the inhibitory and excitatory autapse, which are characterized by changes of bifurcations, frequency responses, steady-state current-potential curves, and nullclines. Furthermore, we identified codimension-1 and -2 bifurcations and the characteristics of the current-potential curve that determine the transitions. Our results presented a comprehensive relationship between 2 classes of neuronal excitability/spiking characterized by different types of bifurcations, along with a novel possible function of autapse or self-feedback control on modulating neuronal excitability.

Software for Brain Network Simulations: A Comparative Study

Numerical simulations of brain networks are a critical part of our efforts in understanding brain functions under pathological and normal conditions. For several decades, the community has developed many software packages and simulators to accelerate research in computational neuroscience. In this article, we select the three most popular simulators, as determined by the number of models in the ModelDB database, such as NEURON, GENESIS, and BRIAN, and perform an independent evaluation of these simulators. In addition, we study NEST, one of the lead simulators of the Human Brain Project. First, we study them based on one of the most important characteristics, the range of supported models. Our investigation reveals that brain network simulators may be biased toward supporting a specific set of models. However, all simulators tend to expand the supported range of models by providing a universal environment for the computational study of individual neurons and brain networks. Next, our investigations on the characteristics of computational architecture and efficiency indicate that all simulators compile the most computationally intensive procedures into binary code, with the aim of maximizing their computational performance. However, not all simulators provide the simplest method for module development and/or guarantee efficient binary code. Third, a study of their amenability for high-performance computing reveals that NEST can almost transparently map an existing model on a cluster or multicore computer, while NEURON requires code modification if the model developed for a single computer has to be mapped on a computational cluster. Interestingly, parallelization is the weakest characteristic of BRIAN, which provides no support for cluster computations and limited support for multicore computers. Fourth, we identify the level of user support and frequency of usage for all simulators. Finally, we carry out an evaluation using two case studies: a large network with simplified neural and synaptic models and a small network with detailed models. These two case studies allow us to avoid any bias toward a particular software package. The results indicate that BRIAN provides the most concise language for both cases considered. Furthermore, as expected, NEST mostly favors large network models, while NEURON is better suited for detailed models. Overall, the case studies reinforce our general observation that simulators have a bias in the computational performance toward specific types of the brain network models.

Mechanisms of generation of membrane potential resonance in a neuron with multiple resonant ionic currents

Neuronal membrane potential resonance (MPR) is associated with subthreshold and network oscillations. A number of voltage-gated ionic currents can contribute to the generation or amplification of MPR, but how the interaction of these currents with linear currents contributes to MPR is not well understood. We explored this in the pacemaker PD neurons of the crab pyloric network. The PD neuron MPR is sensitive to blockers of H- (I_H) and calcium-currents (I_Ca). We used the impedance profile of the biological PD neuron, measured in voltage clamp, to constrain parameter values of a conductance-based model using a genetic algorithm and obtained many optimal parameter combinations. Unlike most cases of MPR, in these optimal models, the values of resonant- (f_res) and phasonant- (f_{ϕ = 0}) frequencies were almost identical. Taking advantage of this fact, we linked the peak phase of ionic currents to their amplitude, in order to provide a mechanistic explanation the dependence of MPR on the I_Ca gating variable time constants. Additionally, we found that distinct pairwise correlations between I_Ca parameters contributed to the maintenance of f_res and resonance power (Q_Z). Measurements of the PD neuron MPR at more hyperpolarized voltages resulted in a reduction of f_res but no change in Q_Z. Constraining the optimal models using these data unmasked a positive correlation between the maximal conductances of I_H and I_Ca. Thus, although I_H is not necessary for MPR in this neuron type, it contributes indirectly by constraining the parameters of I_Ca.

Many neuron types exhibit membrane potential resonance (MPR) in which the neuron produces the largest response to oscillatory input at some preferred (resonant) frequency and, in many systems, the network frequency is correlated with neuronal MPR. MPR is captured by a peak in the impedance vs. frequency curve (Z-profile), which is shaped by the dynamics of voltage-gated ionic currents. Although neuron types can express variable levels of ionic currents, they may have a stable resonant frequency. We used the PD neuron of the crab pyloric network to understand how MPR emerges from the interplay of the biophysical properties of multiple ionic currents, each capable of generating resonance. We show the contribution of an inactivating current at the resonant frequency in terms of interacting time constants. We measured the Z-profile of the PD neuron and explored possible combinations of model parameters that fit this experimentally measured profile. We found that the Z-profile constrains and defines correlations among parameters associated with ionic currents. Furthermore, the resonant frequency and amplitude are sensitive to different parameter sets and can be preserved by co-varying pairs of parameters along their correlation lines. Furthermore, although a resonant current may be present in a neuron, it may not directly contribute to MPR, but constrain the properties of other currents that generate MPR. Finally, constraining model parameters further to those that modify their MPR properties to changes in voltage range produces maximal conductance correlations.

Resonance modulation, annihilation and generation of anti-resonance and anti-phasonance in 3D neuronal systems: interplay of resonant and amplifying currents with slow dynamics

Subthreshold (membrane potential) resonance and phasonance (preferred amplitude and zero-phase responses to oscillatory inputs) in single neurons arise from the interaction between positive and negative feedback effects provided by relatively fast amplifying currents and slower resonant currents. In 2D neuronal systems, amplifying currents are required to be slave to voltage (instantaneously fast) for these phenomena to occur. In higher dimensional systems, additional currents operating at various effective time scales may modulate and annihilate existing resonances and generate antiresonance (minimum amplitude response) and antiphasonance (zero-phase response with phase monotonic properties opposite to phasonance). We use mathematical modeling, numerical simulations and dynamical systems tools to investigate the mechanisms underlying these phenomena in 3D linear models, which are obtained as the linearization of biophysical (conductance-based) models. We characterize the parameter regimes for which the system exhibits the various types of behavior mentioned above in the rather general case in which the underlying 2D system exhibits resonance. We consider two cases: (i) the interplay of two resonant gating variables, and (ii) the interplay of one resonant and one amplifying gating variables. Increasing levels of an amplifying current cause (i) a response amplification if the amplifying current is faster than the resonant current, (ii) resonance and phasonance attenuation and annihilation if the amplifying and resonant currents have identical dynamics, and (iii) antiresonance and antiphasonance if the amplifying current is slower than the resonant current. We investigate the underlying mechanisms by extending the envelope-plane diagram approach developed in previous work (for 2D systems) to three dimensions to include the additional gating variable, and constructing the corresponding envelope curves in these envelope-space diagrams. We find that antiresonance and antiphasonance emerge as the result of an asymptotic boundary layer problem in the frequency domain created by the different balances between the intrinsic time constants of the cell and the input frequency f as it changes. For large enough values of f the envelope curves are quasi-2D and the impedance profile decreases with the input frequency. In contrast, for f ≪ 1 the dynamics are quasi-1D and the impedance profile increases above the limiting value in the other regime. Antiresonance is created because the continuity of the solution requires the impedance profile to connect the portions belonging to the two regimes. If in doing so the phase profile crosses the zero value, then antiphasonance is also generated.

NCAM Regulates Inhibition and Excitability in Layer 2/3 Pyramidal Cells of Anterior Cingulate Cortex

The neural cell adhesion molecule (NCAM), has been shown to be an obligate regulator of synaptic stability and pruning during critical periods of cortical maturation. However, the functional consequences of NCAM deletion on the organization of inhibitory circuits in cortex are not known. In vesicular gamma-amino butyric acid (GABA) transporter (VGAT)-channelrhodopsin2 (ChR2)-enhanced yellow fluorescent protein (EYFP) transgenic mice, NCAM is expressed postnatally at perisomatic synaptic puncta of EYFP-labeled parvalbumin, somatostatin and calretinin-positive interneurons, and in the neuropil in the anterior cingulate cortex (ACC). To investigate how NCAM deletion affects the spatial organization of inhibitory inputs to pyramidal cells, we used laser scanning photostimulation in brain slices of VGAT-ChR2-EYFP transgenic mice crossed to either NCAM-null or wild type (WT) mice. Laser scanning photostimulation revealed that NCAM deletion increased the strength of close-in inhibitory connections to layer 2/3 pyramidal cells of the ACC. In addition, in NCAM-null mice, the intrinsic excitability of pyramidal cells increased, whereas the intrinsic excitability of GABAergic interneurons did not change. The increase in inhibitory tone onto pyramidal cells, and the increased pyramidal cell excitability in NCAM-null mice will alter the delicate coordination of excitation and inhibition (E/I coordination) in the ACC, and may be a factor contributing to circuit dysfunction in diseases such as schizophrenia and bipolar disorder, in which NCAM has been implicated.

Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

The synchronization tendencies of networks of oscillators have been studied intensely. We assume a network of all-to-all pulse-coupled oscillators in which the effect of a pulse is independent of the number of oscillators that simultaneously emit a pulse and the normalized delay (the phase resetting) is a monotonically increasing function of oscillator phase with the slope everywhere less than 1 and a value greater than 2φ-1, where φ is the normalized phase. Order switching cannot occur; the only possible solutions are globally attracting synchrony and cluster solutions with a fixed firing order. For small conduction delays, we prove the former stable and all other possible attractors nonexistent due to the destabilizing discontinuity of the phase resetting at a phase of 0.

Modeling fast and slow gamma oscillations with interneurons of different subtype

Experimental and theoretical studies demonstrate that neuronal gamma oscillations crucially depend on interneurons, but current models do not consider the diversity of known interneuron subtypes. Moreover, in CA1 of the hippocampus, experimental evidence indicates the presence of multiple gamma oscillators, two of which may be coordinated by differing interneuron populations. In this article, we show that models of networks with competing interneuron populations with different postsynaptic effects are sufficient to generate, within CA1, distinct oscillatory regimes. We find that strong mutual inhibition between the interneuron populations permits distinct fast and slow gamma states, whereas weak mutual inhibition generates mixed gamma states. We develop idealized firing rate models to illuminate dynamic properties of these competitive gamma networks, and reinforce these concepts with basic spiking models. The models make several explicit predictions about gamma oscillators in CA1. Specifically, interneurons of different subtype phase-lock to different gamma states, and one population of interneurons is silenced and the other active during fast and slow gamma events. Finally, mutual inhibition between interneuron populations is necessary to generate distinct gamma states. Previous experimental studies indicate that fast and slow gamma oscillations reflect different information processing modes, although it is unclear whether these rhythms are intrinsic or imposed. The models outlined demonstrate that basic architectures can locally generate these oscillations, as well as capture other features of fast and slow gamma, including theta-phase preference and spontaneous transitions between gamma states. These models may extend to describe general dynamics in networks with diverse interneuron populations.NEW & NOTEWORTHY The oscillatory coordination of neural signals is crucial to healthy brain function. We have developed an idealized neuronal model that generates distinct fast and slow gamma oscillations, a known feature of the rodent hippocampus. Our work provides a mechanism of this phenomenon, as well as a theoretical framework for future experiments concerning hippocampal gamma. It moreover offers a tractable model of competitive gamma oscillations that is generalizable across the nervous system.

The shaping of intrinsic membrane potential oscillations: positive/negative feedback, ionic resonance/amplification, nonlinearities and time scales

The generation of intrinsic subthreshold (membrane potential) oscillations (STOs) in neuronal models requires the interaction between two processes: a relatively fast positive feedback that favors changes in voltage and a slower negative feedback that opposes these changes. These are provided by the so-called resonant and amplifying gating variables associated to the participating ionic currents. We investigate both the biophysical and dynamic mechanisms of generation of STOs and how their attributes (frequency and amplitude) depend on the model parameters for biophysical (conductance-based) models having qualitatively different types of resonant currents (activating and inactivating) and an amplifying current. Combinations of the same types of ionic currents (same models) in different parameter regimes give rise to different types of nonlinearities in the voltage equation: quasi-linear, parabolic-like and cubic-like. On the other hand, combinations of different types of ionic currents (different models) may give rise to the same type of nonlinearities. We examine how the attributes of the resulting STOs depend on the combined effect of these resonant and amplifying ionic processes, operating at different effective time scales, and the various types of nonlinearities. We find that, while some STO properties and attribute dependencies on the model parameters are determined by the specific combinations of ionic currents (biophysical properties), and are different for models with different such combinations, others are determined by the type of nonlinearities and are common for models with different types of ionic currents. Our results highlight the richness of STO behavior in single cells as the result of the various ways in which resonant and amplifying currents interact and affect the generation and termination of STOs as control parameters change. We make predictions that can be tested experimentally and are expected to contribute to the understanding of how rhythmic activity in neuronal networks emerge from the interplay of the intrinsic properties of the participating neurons and the network connectivity.

Intrinsic Cellular Properties and Connectivity Density Determine Variable Clustering Patterns in Randomly Connected Inhibitory Neural Networks

The plethora of inhibitory interneurons in the hippocampus and cortex play a pivotal role in generating rhythmic activity by clustering and synchronizing cell firing. Results of our simulations demonstrate that both the intrinsic cellular properties of neurons and the degree of network connectivity affect the characteristics of clustered dynamics exhibited in randomly connected, heterogeneous inhibitory networks. We quantify intrinsic cellular properties by the neuron's current-frequency relation (IF curve) and Phase Response Curve (PRC), a measure of how perturbations given at various phases of a neurons firing cycle affect subsequent spike timing. We analyze network bursting properties of networks of neurons with Type I or Type II properties in both excitability and PRC profile; Type I PRCs strictly show phase advances and IF curves that exhibit frequencies arbitrarily close to zero at firing threshold while Type II PRCs display both phase advances and delays and IF curves that have a non-zero frequency at threshold. Type II neurons whose properties arise with or without an M-type adaptation current are considered. We analyze network dynamics under different levels of cellular heterogeneity and as intrinsic cellular firing frequency and the time scale of decay of synaptic inhibition are varied. Many of the dynamics exhibited by these networks diverge from the predictions of the interneuron network gamma (ING) mechanism, as well as from results in all-to-all connected networks. Our results show that randomly connected networks of Type I neurons synchronize into a single cluster of active neurons while networks of Type II neurons organize into two mutually exclusive clusters segregated by the cells' intrinsic firing frequencies. Networks of Type II neurons containing the adaptation current behave similarly to networks of either Type I or Type II neurons depending on network parameters; however, the adaptation current creates differences in the cluster dynamics compared to those in networks of Type I or Type II neurons. To understand these results, we compute neuronal PRCs calculated with a perturbation matching the profile of the synaptic current in our networks. Differences in profiles of these PRCs across the different neuron types reveal mechanisms underlying the divergent network dynamics.

Slow gamma rhythms in CA3 are entrained by slow gamma activity in the dentate gyrus

In hippocampal area CA1, slow (∼25-55 Hz) and fast (∼60-100 Hz) gamma rhythms are coupled with different CA1 afferents. CA1 slow gamma is coupled to inputs from CA3, and CA1 fast gamma is coupled to inputs from the medial entorhinal cortex (Colgin LL, Denninger T, Fyhn M, Hafting T, Bonnevie T, Jensen O, Moser MB, Moser EI. Nature 462: 353-357, 2009). CA3 gives rise to highly divergent associational projections, and it is possible that reverberating activity in these connections generates slow gamma rhythms in the hippocampus. However, hippocampal gamma is maximal upstream of CA3, in the dentate gyrus (DG) region (Bragin A, Jando G, Nadasdy Z, Hetke J, Wise K, Buzsaki G. J Neurosci 15: 47-60, 1995). Thus it is possible that slow gamma in CA3 is driven by inputs from DG, yet few studies have examined slow and fast gamma rhythms in DG recordings. Here we investigated slow and fast gamma rhythms in paired recordings from DG and CA3 in freely moving rats to determine whether slow and fast gamma rhythms in CA3 are entrained by DG. We found that slow gamma rhythms, as opposed to fast gamma rhythms, were particularly prominent in DG. We investigated directional causal influences between DG and CA3 by Granger causality analysis and found that DG slow gamma influences CA3 slow gamma. Moreover, DG place cell spikes were strongly phase-locked to CA3 slow gamma rhythms, suggesting that DG excitatory projections to CA3 may underlie this directional influence. These results indicate that slow gamma rhythms do not originate in CA3 but rather slow gamma activity upstream in DG entrains slow gamma rhythms in CA3.

Membrane potential resonance frequency directly influences network frequency through electrical coupling

Oscillatory networks often include neurons with membrane potential resonance, exhibiting a peak in the voltage amplitude as a function of current input at a nonzero (resonance) frequency (f_res). Although f_res has been correlated to the network frequency (f_net) in a variety of systems, a causal relationship between the two has not been established. We examine the hypothesis that combinations of biophysical parameters that shift f_res, without changing other attributes of the impedance profile, also shift f_net in the same direction. We test this hypothesis, computationally and experimentally, in an electrically coupled network consisting of intrinsic oscillator (O) and resonator (R) neurons. We use a two-cell model of such a network to show that increasing f_res of R directly increases f_net and that this effect becomes more prominent if the amplitude of resonance is increased. Notably, the effect of f_res on f_net is independent of the parameters that define the oscillator or the combination of parameters in R that produce the shift in f_res, as long as this combination produces the same impedance vs. frequency relationship. We use the dynamic clamp technique to experimentally verify the model predictions by connecting a model resonator to the pacemaker pyloric dilator neurons of the crab Cancer borealis pyloric network using electrical synapses and show that the pyloric network frequency can be shifted by changing f_res in the resonator. Our results provide compelling evidence that f_res and resonance amplitude strongly influence f_net, and therefore, modulators may target these attributes to modify rhythmic activity.