Resonance or Integration? Self-Sustained Dynamics and Excitability of Neural Microcircuits

The fractal brain: scale-invariance in structure and dynamics

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

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

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

Recurrent dynamics in the cerebral cortex: Integration of sensory evidence with stored knowledge

Current concepts of sensory processing in the cerebral cortex emphasize serial extraction and recombination of features in hierarchically structured feed-forward networks in order to capture the relations among the components of perceptual objects. These concepts are implemented in convolutional deep learning networks and have been validated by the astounding similarities between the functional properties of artificial systems and their natural counterparts. However, cortical architectures also display an abundance of recurrent coupling within and between the layers of the processing hierarchy. This massive recurrence gives rise to highly complex dynamics whose putative function is poorly understood. Here a concept is proposed that assigns specific functions to the dynamics of cortical networks and combines, in a unifying approach, the respective advantages of recurrent and feed-forward processing. It is proposed that the priors about regularities of the world are stored in the weight distributions of feed-forward and recurrent connections and that the high-dimensional, dynamic space provided by recurrent interactions is exploited for computations. These comprise the ultrafast matching of sensory evidence with the priors covertly represented in the correlation structure of spontaneous activity and the context-dependent grouping of feature constellations characterizing natural objects. The concept posits that information is encoded not only in the discharge frequency of neurons but also in the precise timing relations among the discharges. Results of experiments designed to test the predictions derived from this concept support the hypothesis that cerebral cortex exploits the high-dimensional recurrent dynamics for computations serving predictive coding.

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

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

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).

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.

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.

Topological determinants of self-sustained activity in a simple model of excitable dynamics on graphs

Simple models of excitable dynamics on graphs are an efficient framework for studying the interplay between network topology and dynamics. This topic is of practical relevance to diverse fields, ranging from neuroscience to engineering. Here we analyze how a single excitation propagates through a random network as a function of the excitation threshold, that is, the relative amount of activity in the neighborhood required for the excitation of a node. We observe that two sharp transitions delineate a region of sustained activity. Using analytical considerations and numerical simulation, we show that these transitions originate from the presence of barriers to propagation and the excitation of topological cycles, respectively, and can be predicted from the network topology. Our findings are interpreted in the context of network reverberations and self-sustained activity in neural systems, which is a question of long-standing interest in computational neuroscience.

Complete Firing-Rate Response of Neurons with Complex Intrinsic Dynamics

The response of a neuronal population over a space of inputs depends on the intrinsic properties of its constituent neurons. Two main modes of single neuron dynamics–integration and resonance–have been distinguished. While resonator cell types exist in a variety of brain areas, few models incorporate this feature and fewer have investigated its effects. To understand better how a resonator’s frequency preference emerges from its intrinsic dynamics and contributes to its local area’s population firing rate dynamics, we analyze the dynamic gain of an analytically solvable two-degree of freedom neuron model. In the Fokker-Planck approach, the dynamic gain is intractable. The alternative Gauss-Rice approach lifts the resetting of the voltage after a spike. This allows us to derive a complete expression for the dynamic gain of a resonator neuron model in terms of a cascade of filters on the input. We find six distinct response types and use them to fully characterize the routes to resonance across all values of the relevant timescales. We find that resonance arises primarily due to slow adaptation with an intrinsic frequency acting to sharpen and adjust the location of the resonant peak. We determine the parameter regions for the existence of an intrinsic frequency and for subthreshold and spiking resonance, finding all possible intersections of the three. The expressions and analysis presented here provide an account of how intrinsic neuron dynamics shape dynamic population response properties and can facilitate the construction of an exact theory of correlations and stability of population activity in networks containing populations of resonator neurons.

Dynamic gain, the amount by which features at specific frequencies in the input to a neuron are amplified or attenuated in its output spiking, is fundamental for the encoding of information by neural populations. Most studies of dynamic gain have focused on neurons without intrinsic degrees of freedom exhibiting integrator-type subthreshold dynamics. Many neuron types in the brain, however, exhibit complex subthreshold dynamics such as resonance, found for instance in cortical interneurons, stellate cells, and mitral cells. A resonator neuron has at least two degrees of freedom for which the classical Fokker-Planck approach to calculating the dynamic gain is largely intractable. Here, we lift the voltage-reset rule after a spike, allowing us to derive a complete expression of the dynamic gain of a resonator neuron model. We find the gain can exhibit only six shapes. The resonant ones have peaks that become large due to intrinsic adaptation and become sharp due to an intrinsic frequency. A resonance can nevertheless result from either property. The analysis presented here helps explain how intrinsic neuron dynamics shape population-level response properties and provides a powerful tool for developing theories of inter-neuron correlations and dynamic responses of neural populations.

Formation and disruption of tonotopy in a large-scale model of the auditory cortex

There is ample experimental evidence describing changes of tonotopic organisation in the auditory cortex due to environmental factors. In order to uncover the underlying mechanisms, we designed a large-scale computational model of the auditory cortex. The model has up to 100 000 Izhikevich's spiking neurons of 17 different types, almost 21 million synapses, which are evolved according to Spike-Timing-Dependent Plasticity (STDP) and have an architecture akin to existing observations. Validation of the model revealed alternating synchronised/desynchronised states and different modes of oscillatory activity. We provide insight into these phenomena via analysing the activity of neuronal subtypes and testing different causal interventions into the simulation. Our model is able to produce experimental predictions on a cell type basis. To study the influence of environmental factors on the tonotopy, different types of auditory stimulations during the evolution of the network were modelled and compared. We found that strong white noise resulted in completely disrupted tonotopy, which is consistent with in vivo experimental observations. Stimulation with pure tones or spontaneous activity led to a similar degree of tonotopy as in the initial state of the network. Interestingly, weak white noise led to a substantial increase in tonotopy. As the STDP was the only mechanism of plasticity in our model, our results suggest that STDP is a sufficient condition for the emergence and disruption of tonotopy under various types of stimuli. The presented large-scale model of the auditory cortex and the core simulator, SUSNOIMAC, have been made publicly available.

Subthreshold amplitude and phase resonance in models of quadratic type: nonlinear effects generated by the interplay of resonant and amplifying currents

We investigate the biophysical and dynamic mechanisms of generation of subthreshold amplitude and phase resonance in response to sinusoidal input currents in two-dimensional models of quadratic type. These models feature a parabolic voltage nullcline and a linear nullcline for the recovery gating variable, capturing the interplay of the so-called resonant currents (e.g., hyperpolarization-activated mixed-cation inward and slow potassium) and amplifying currents (e.g., persistent sodium) in biophysically realistic parameter regimes. These currents underlie the generation of resonance in medial entorhinal cortex layer II stellate cells and CA1 pyramidal cells. We show that quadratic models exhibit nonlinear amplifications of the voltage response to sinusoidal inputs in the resonant frequency band. These are expressed as an increase in the impedance profile as the input amplitude increases. They are stronger for values positive than negative to resting potential and are accompanied by a shift in the phase profile, a decrease in the resonant and phase-resonant frequencies, and an increase in the sharpness of the voltage response. These effects are more prominent for smaller values of ∊ (larger levels of the time scale separation between the voltage and the resonant gating variable) and for values of the resting potential closer to threshold for spike generation. All other parameter fixed, as ∊ increases the voltage response becomes "more linear"; i.e., the nonlinearities are present, but "ignored". In addition, the nonlinear effects are strongly modulated by the curvature of the parabolic voltage nullcline (partially reflecting the effects of the amplifying current) and the slope of the resonant current activation curve. Following the effects of changes in the biophysical conductances of realistic conductance-based models through the parameters of the quadratic model, we characterize the qualitatively different effects that resonant and amplifying currents have on the nonlinear properties of the voltage response. We identify different classes of resonant currents, represented by h- and slow potassium, according to whether they enhance (h-) or attenuate (slow potassium) the nonlinear effects. Finally, we use dynamical systems tools to investigate the dynamic mechanisms of generation of resonance and phase-resonance. We show that the nonlinear effects on the voltage response (e.g., amplification of the voltage response in the resonant frequency band and shifts in the resonant and phase-resonant frequencies) result from the ability of limit cycle trajectories to follow the unstable (right) branch of the voltage nullcline for a significant amount of time. This is a canard-related mechanism that has been shown to underlie the generation of intrinsic subthreshold oscillations in quadratic type models such as medial entorhinal cortex stellate cells. Overall, our results highlight the complexity of the voltage response to oscillatory inputs in nonlinear models and the roles that resonant and amplifying currents have in shaping these responses.

Neuronal oscillations in Golgi cells and Purkinje cells are accompanied by decreases in Shannon information entropy

Neuronal oscillations have been shown to contribute to the function of the cerebral cortex by coordinating the neuronal activities of distant cortical regions via a temporal synchronization of neuronal discharge patterns. This can occur regardless whether these regions are linked by cortico-cortical pathways or not. Less is known concerning the role of neuronal oscillations in the cerebellum. Golgi cells and Purkinje cells are both principal cell types in the cerebellum. Purkinje cells are the sole output cells of the cerebellar cortex while Golgi cells contribute to information processing at the input stage of the cerebellar cortex. Both cell types have large cell bodies, as well as dendritic structures, that can generate large currents. The discharge patterns of both these cell types also exhibit oscillations. In view of the massive afferent information conveyed by the mossy fiber-granule cell system to different and distant areas of the cerebellar cortex, it is relevant to inquire the role of cerebellar neuronal oscillations in information processing. In this study, we compared the discharge patterns of Golgi cells and Purkinje cells in conscious rats and in rats anesthetized with urethane. We assessed neuronal oscillations by analyzing the regularity in the timing of individual spikes within a spike train by using autocorrelograms and fast-Fourier transform. We measured the differences in neuronal oscillations and the amount of information content in a spike train (defined by Shannon entropy processed per unit time) in rats under anesthesia and in conscious, awake rats. Our findings indicated that anesthesia caused more prominent neuronal oscillations in both Golgi cells and Purkinje cells accompanied by decreases in Shannon information entropy in their spike trains.

Membrane resonance enables stable and robust gamma oscillations

Neuronal mechanisms underlying beta/gamma oscillations (20–80 Hz) are not completely understood. Here, we show that in vivo beta/gamma oscillations in the cat visual cortex sometimes exhibit remarkably stable frequency even when inputs fluctuate dramatically. Enhanced frequency stability is associated with stronger oscillations measured in individual units and larger power in the local field potential. Simulations of neuronal circuitry demonstrate that membrane properties of inhibitory interneurons strongly determine the characteristics of emergent oscillations. Exploration of networks containing either integrator or resonator inhibitory interneurons revealed that: (i) Resonance, as opposed to integration, promotes robust oscillations with large power and stable frequency via a mechanism called RING (Resonance INduced Gamma); resonance favors synchronization by reducing phase delays between interneurons and imposes bounds on oscillation cycle duration; (ii) Stability of frequency and robustness of the oscillation also depend on the relative timing of excitatory and inhibitory volleys within the oscillation cycle; (iii) RING can reproduce characteristics of both Pyramidal INterneuron Gamma (PING) and INterneuron Gamma (ING), transcending such classifications; (iv) In RING, robust gamma oscillations are promoted by slow but are impaired by fast inputs. Results suggest that interneuronal membrane resonance can be an important ingredient for generation of robust gamma oscillations having stable frequency.

An insect-inspired bionic sensor for tactile localization and material classification with state-dependent modulation

Insects carry a pair of antennae on their head: multimodal sensory organs that serve a wide range of sensory-guided behaviors. During locomotion, antennae are involved in near-range orientation, for example in detecting, localizing, probing, and negotiating obstacles. Here we present a bionic, active tactile sensing system inspired by insect antennae. It comprises an actuated elastic rod equipped with a terminal acceleration sensor. The measurement principle is based on the analysis of damped harmonic oscillations registered upon contact with an object. The dominant frequency of the oscillation is extracted to determine the distance of the contact point along the probe and basal angular encoders allow tactile localization in a polar coordinate system. Finally, the damping behavior of the registered signal is exploited to determine the most likely material. The tactile sensor is tested in four approaches with increasing neural plausibility: first, we show that peak extraction from the Fourier spectrum is sufficient for tactile localization with position errors below 1%. Also, the damping property of the extracted frequency is used for material classification. Second, we show that the Fourier spectrum can be analysed by an Artificial Neural Network (ANN) which can be trained to decode contact distance and to classify contact materials. Thirdly, we show how efficiency can be improved by band-pass filtering the Fourier spectrum by application of non-negative matrix factorization. This reduces the input dimension by 95% while reducing classification performance by 8% only. Finally, we replace the FFT by an array of spiking neurons with gradually differing resonance properties, such that their spike rate is a function of the input frequency. We show that this network can be applied to detect tactile contact events of a wheeled robot, and how detrimental effects of robot velocity on antennal dynamics can be suppressed by state-dependent modulation of the input signals.

Visualization for understanding of neurodynamical systems

Complex neurodynamical systems are quite difficult to analyze and understand. New type of plots are introduced to help in visualization of high-dimensional trajectories and show global picture of the phase space, including relations between basins of attractors. Color recurrence plots (RPs) display distances from each point on the trajectory to all other points in a two-dimensional matrix. Fuzzy Symbolic Dynamics (FSD) plots enhance this information mapping the whole trajectory to two or three dimensions. Each coordinate is defined by the value of a fuzzy localized membership function, optimized to visualize interesting features of the dynamics, showing to which degree a point on the trajectory belongs to some neighborhood. The variance of the trajectory within the attraction basin plotted against the variance of the synaptic noise provides information about sizes and shapes of these basins. Plots that use color to show the distance between each trajectory point and a larger number of selected reference points (for example centers of attractor basins) are also introduced. Activity of 140 neurons in the semantic layer of dyslexia model implemented in the Emergent neural simulator is analyzed in details showing different aspects of neurodynamics that may be understood in this way. Influence of connectivity and various neural properties on network dynamics is illustrated using visualization techniques. A number of interesting conclusions about cognitive neurodynamics of lexical concept activations are drawn. Changing neural accommodation parameters has very strong influence on the dwell time of the trajectories. This may be linked to attention deficits disorders observed in autism in case of strong enslavement, and to ADHD-like behavior in case of weak enslavement.

Inferring single neuron properties in conductance based balanced networks

Balanced states in large networks are a usual hypothesis for explaining the variability of neural activity in cortical systems. In this regime the statistics of the inputs is characterized by static and dynamic fluctuations. The dynamic fluctuations have a Gaussian distribution. Such statistics allows to use reverse correlation methods, by recording synaptic inputs and the spike trains of ongoing spontaneous activity without any additional input. By using this method, properties of the single neuron dynamics that are masked by the balanced state can be quantified. To show the feasibility of this approach we apply it to large networks of conductance based neurons. The networks are classified as Type I or Type II according to the bifurcations which neurons of the different populations undergo near the firing onset. We also analyze mixed networks, in which each population has a mixture of different neuronal types. We determine under which conditions the intrinsic noise generated by the network can be used to apply reverse correlation methods. We find that under realistic conditions we can ascertain with low error the types of neurons present in the network. We also find that data from neurons with similar firing rates can be combined to perform covariance analysis. We compare the results of these methods (that do not requite any external input) to the standard procedure (that requires the injection of Gaussian noise into a single neuron). We find a good agreement between the two procedures.

Timescales of multineuronal activity patterns reflect temporal structure of visual stimuli

The investigation of distributed coding across multiple neurons in the cortex remains to this date a challenge. Our current understanding of collective encoding of information and the relevant timescales is still limited. Most results are restricted to disparate timescales, focused on either very fast, e.g., spike-synchrony, or slow timescales, e.g., firing rate. Here, we investigated systematically multineuronal activity patterns evolving on different timescales, spanning the whole range from spike-synchrony to mean firing rate. Using multi-electrode recordings from cat visual cortex, we show that cortical responses can be described as trajectories in a high-dimensional pattern space. Patterns evolve on a continuum of coexisting timescales that strongly relate to the temporal properties of stimuli. Timescales consistent with the time constants of neuronal membranes and fast synaptic transmission (5–20 ms) play a particularly salient role in encoding a large amount of stimulus-related information. Thus, to faithfully encode the properties of visual stimuli the brain engages multiple neurons into activity patterns evolving on multiple timescales.

Neurophysiological and computational principles of cortical rhythms in cognition

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

Optimal hierarchical modular topologies for producing limited sustained activation of neural networks

An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable regimes of network activation, typically arising from a limited parameter range. In this range of limited sustained activity (LSA), the activity of neural populations in the network persists between the extremes of either quickly dying out or activating the whole network. Hierarchical modular networks were previously found to show a wider parameter range for LSA than random or small-world networks not possessing hierarchical organization or multiple modules. Here we explored how variation in the number of hierarchical levels and modules per level influenced network dynamics and occurrence of LSA. We tested hierarchical configurations of different network sizes, approximating the large-scale networks linking cortical columns in one hemisphere of the rat, cat, or macaque monkey brain. Scaling of the network size affected the number of hierarchical levels and modules in the optimal networks, also depending on whether global edge density or the numbers of connections per node were kept constant. For constant edge density, only few network configurations, possessing an intermediate number of levels and a large number of modules, led to a large range of LSA independent of brain size. For a constant number of node connections, there was a trend for optimal configurations in larger-size networks to possess a larger number of hierarchical levels or more modules. These results may help to explain the trend to greater network complexity apparent in larger brains and may indicate that this complexity is required for maintaining stable levels of neural activation.

Self-sustained irregular activity in 2-D small-world networks of excitatory and inhibitory neurons

In this paper, we study the self-sustained irregular firing activity in 2-D small-world (SW) neural networks consisting of both excitatory and inhibitory neurons by computational modeling. For a proper proportion of unidirectional shortcuts, the stable self-sustained activity with irregular firing states indeed occurs in the considered network. By varying the shortcut density while keeping other system parameters fixed, different levels of irregular firing states, from weakly irregular to Poisson-like and burst firing states, are obtained in 2-D SW neural networks. It is also observed that this activity is sensitive to small perturbations, which might provide a possible mechanism for producing chaos. On the other hand, we find that several other system parameters, such as the network size and refractory period, have significant impact on this activity. Further simulation results show that the 2-D SW neural network can sustain such long-lasting firing behavior by using a smaller number of connections than the random neural network.

Spike timing-dependent plasticity is affected by the interplay of intrinsic and network oscillations

Spike timing-dependent plasticity (STDP) is a form of Hebbian learning which is thought to underlie structure formation during development, and learning and memory in later life. In this paper we show that the intrinsic properties of the postsynaptic neuron might have a deep influence on STDP dynamics by shaping the causal correlation between the pre- and the postsynaptic spike trains. The cell-specific effect of STDP is particularly evident in the presence of an oscillatory component in a cell input. In this case, the cell-specific phase response to an oscillatory modulation biases the oscillating afferents towards potentiation or depression, depending upon the intrinsic dynamics of the postsynaptic neuron and the period of the modulation.

Response of a Hodgkin-Huxley neuron to a high-frequency input

We study the response of a Hodgkin-Huxley neuron stimulated by a periodic sequence of conductance pulses arriving through the synapse in the high-frequency regime. In addition to the usual excitation threshold there is a smooth crossover from the firing to the silent regime for increasing pulse amplitude gsyn. The amplitude of the voltage spikes decreases approximately linearly with gsyn. In some regions of parameter space the response is irregular, probably chaotic. In the chaotic regime between the mode-locked regions 3:1 and 2:1 near the lower excitation threshold, the output interspike interval histogram (ISIH) undergoes a sharp transition. If the driving period is below the critical value, T<T*, the output histogram contains only odd multiples of Ti. For Ti>T* even multiples of Ti also appear in the histogram, starting from the largest values. Near T* the ISIH scales logarithmically on both sides of the transition. The coefficient of variation of ISIH has a cusp singularity at T*. The average response period has a maximum slightly above T*. Near the excitation threshold in the chaotic regime the average firing rate rises sublinearly from frequencies of order 1 Hz.

A color-based visualization technique for multielectrode spike trains

Multi electrode recordings of neuronal activity provide an overwhelming amount of data that is often difficult to analyze and interpret. Although various methods exist for treating multielectrode datasets quantitatively, there is a particularly prominent lack of techniques that enable a quick visual exploration of such datasets. Here, by using Kohonen self-organizing maps, we propose a simple technique that allows for the representation of multiple spike trains through a sequence of color-coded population activity vectors. When multiple color sequences are grouped according to a certain criterion, e.g., by stimulation condition or recording time, one can inspect an entire dataset visually and extract quickly information about the identity, stimulus-locking and temporal distribution of multi-neuron activity patterns. Color sequences can be computed on various time scales revealing different aspects of the temporal dynamics and can emphasize high-order correlation patterns that are not detectable with pairwise techniques. Furthermore, this technique is useful for determining the stability of neuronal responses during a recording session. Due to its simplicity and reliance on perceptual grouping, the method is useful for both quick on-line visualization of incoming data and for more detailed post hoc analyses.

Receptive Field Self-Organization in a Model of the Fine Structure in V1 Cortical Columns

We study a dynamical model of processing and learning in the visual cortex, which reflects the anatomy of V1 cortical columns and properties of their neuronal receptive fields. Based on recent results on the fine-scale structure of columns in V1, we model the activity dynamics in subpopulations of excitatory neurons and their interaction with systems of inhibitory neurons. We find that a dynamical model based on these aspects of columnar anatomy can give rise to specific types of computations that result in self-organization of afferents to the column. For a given type of input, self-organization reliably extracts the basic input components represented by neuronal receptive fields. Self-organization is very noise tolerant and can robustly be applied to different types of input. To quantitatively analyze the system's component extraction capabilities, we use two standard benchmarks: the bars test and natural images. In the bars test, the system shows the highest noise robustness reported so far. If natural image patches are used as input, self-organization results in Gabor-like receptive fields. In quantitative comparison with in vivo measurements, we find that the obtained receptive fields capture statistical properties of V1 simple cells that algorithms such as independent component analysis or sparse coding do not reproduce.

Membrane resonance in bursting pacemaker neurons of an oscillatory network is correlated with network frequency

Network oscillations typically span a limited range of frequency. In pacemaker-driven networks, including many central pattern generators (CPGs), this frequency range is determined by the properties of bursting pacemaker neurons and their synaptic connections; thus, factors that affect the burst frequency of pacemaker neurons should play a role in determining the network frequency. We examine the role of membrane resonance of pacemaker neurons on the network frequency in the crab pyloric CPG. The pyloric oscillations (frequency of approximately 1 Hz) are generated by a group of pacemaker neurons: the anterior burster (AB) and the pyloric dilator (PD). We examine the impedance profiles of the AB and PD neurons in response to sinusoidal current injections with varying frequency and find that both neuron types exhibit membrane resonance, i.e., demonstrate maximal impedance at a given preferred frequency. The membrane resonance frequencies of the AB and PD neurons fall within the range of the pyloric network oscillation frequency. Experiments with pharmacological blockers and computational modeling show that both calcium currents I(Ca) and the hyperpolarization-activated inward current I(h) are important in producing the membrane resonance in these neurons. We then demonstrate that both the membrane resonance frequency of the PD neuron and its suprathreshold bursting frequency can be shifted in the same direction by either direct current injection or by using the dynamic-clamp technique to inject artificial conductances for I(h) or I(Ca). Together, these results suggest that membrane resonance of pacemaker neurons can be strongly correlated with the CPG oscillation frequency.

Working memory dynamics and spontaneous activity in a flip-flop oscillations network model with a Milnor attractor

Many cognitive tasks require the ability to maintain and manipulate simultaneously several chunks of information. Numerous neurobiological observations have reported that this ability, known as the working memory, is associated with both a slow oscillation (leading to the up and down states) and the presence of the theta rhythm. Furthermore, during resting state, the spontaneous activity of the cortex exhibits exquisite spatiotemporal patterns sharing similar features with the ones observed during specific memory tasks. Here to enlighten neural implication of working memory under these complicated dynamics, we propose a phenomenological network model with biologically plausible neural dynamics and recurrent connections. Each unit embeds an internal oscillation at the theta rhythm which can be triggered during up-state of the membrane potential. As a result, the resting state of a single unit is no longer a classical fixed point attractor but rather the Milnor attractor, and multiple oscillations appear in the dynamics of a coupled system. In conclusion, the interplay between the up and down states and theta rhythm endows high potential in working memory operation associated with complexity in spontaneous activities.

Evidence for criterion shifts in visual perceptual learning: data and implications

Work on visual perceptual learning for contrast detection has shown that reliable decreases in detection thresholds are accompanied by reliable increases in false alarm rates (Wenger & Rasche, 2006). The present study assesses the robustness and replicability of these changes, demonstrating that they are independent of a variety of task demands (i.e., the specific method used for perceptual practice and threshold estimation) and the presence or absence of trial-by-trial feedback and that the source of the increases can be found in shifts in changes in sensitivity and in bias for detection, identification, or both. Although the increase in false alarm rates suggests a strategic shift in response criteria for detection, we demonstrate that there are multiple potential explanations, including explanations that do not require strategic shifts on the part of the observer. The empirical evidence and analysis of alternative explanations reinforce the inference that visual perceptual learning may involve more than changes in perceptual sensitivity and that cortical circuits beyond the primary visual areas may be involved.

Type I and Type II Neuron Models Are Selectively Driven by Differential Stimulus Features

Neurons in the nervous system exhibit an outstanding variety of morphological and physiological properties. However, close to threshold, this remarkable richness may be grouped succinctly into two basic types of excitability, often referred to as type I and type II. The dynamical traits of these two neuron types have been extensively characterized. It would be interesting, however, to understand the information-processing consequences of their dynamical properties. To that end, here we determine the differences between the stimulus features inducing firing in type I and type II neurons. We work with both realistic conductance-based models and minimal normal forms. We conclude that type I neurons fire in response to scale-free depolarizing stimuli. Type II neurons, instead, are most efficiently driven by input stimuli containing both depolarizing and hyperpolarizing phases, with significant power in the frequency band corresponding to the intrinsic frequencies of the cell.

Large-scale model of mammalian thalamocortical systems

The understanding of the structural and dynamic complexity of mammalian brains is greatly facilitated by computer simulations. We present here a detailed large-scale thalamocortical model based on experimental measures in several mammalian species. The model spans three anatomical scales. (i) It is based on global (white-matter) thalamocortical anatomy obtained by means of diffusion tensor imaging (DTI) of a human brain. (ii) It includes multiple thalamic nuclei and six-layered cortical microcircuitry based on in vitro labeling and three-dimensional reconstruction of single neurons of cat visual cortex. (iii) It has 22 basic types of neurons with appropriate laminar distribution of their branching dendritic trees. The model simulates one million multicompartmental spiking neurons calibrated to reproduce known types of responses recorded in vitro in rats. It has almost half a billion synapses with appropriate receptor kinetics, short-term plasticity, and long-term dendritic spike-timing-dependent synaptic plasticity (dendritic STDP). The model exhibits behavioral regimes of normal brain activity that were not explicitly built-in but emerged spontaneously as the result of interactions among anatomical and dynamic processes. We describe spontaneous activity, sensitivity to changes in individual neurons, emergence of waves and rhythms, and functional connectivity on different scales.

The oscillation score: an efficient method for estimating oscillation strength in neuronal activity

We present a method that estimates the strength of neuronal oscillations at the cellular level, relying on autocorrelation histograms computed on spike trains. The method delivers a number, termed oscillation score, that estimates the degree to which a neuron is oscillating in a given frequency band. Moreover, it can also reliably identify the oscillation frequency and strength in the given band, independently of the oscillation in other frequency bands, and thus it can handle superimposed oscillations on multiple scales (theta, alpha, beta, gamma, etc.). The method is relatively simple and fast. It can cope with a low number of spikes, converging exponentially fast with the number of spikes, to a stable estimation of the oscillation strength. It thus lends itself to the analysis of spike-sorted single-unit activity from electrophysiological recordings. We show that the method performs well on experimental data recorded from cat visual cortex and also compares favorably to other methods. In addition, we provide a measure, termed confidence score, that determines the stability of the oscillation score estimate over trials.