Dynamical response properties of neocortical neuron ensembles: multiplicative versus additive noise

The steady state and response to a periodic stimulation of the firing rate for a theta neuron with correlated noise

The stochastic activity of neurons is caused by various sources of correlated fluctuations and can be described in terms of simplified, yet biophysically grounded, integrate-and-fire models. One paradigmatic model is the quadratic integrate-and-fire model and its equivalent phase description by the theta neuron. Here we study the theta neuron model driven by a correlated Ornstein-Uhlenbeck noise and by periodic stimuli. We apply the matrix-continued-fraction method to the associated Fokker-Planck equation to develop an efficient numerical scheme to determine the stationary firing rate as well as the stimulus-induced modulation of the instantaneous firing rate. For the stationary case, we identify the conditions under which the firing rate decreases or increases by the effect of the colored noise and compare our results to existing analytical approximations for limit cases. For an additional periodic signal we demonstrate how the linear and nonlinear response terms can be computed and report resonant behavior for some of them. We extend the method to the case of two periodic signals, generally with incommensurable frequencies, and present a particular case for which a strong mixed response to both signals is observed, i.e. where the response to the sum of signals differs significantly from the sum of responses to the single signals. We provide Python code for our computational method: https://github.com/jannikfranzen/theta_neuron .

Ultrafast population coding and axo-somatic compartmentalization

Populations of cortical neurons respond to common input within a millisecond. Morphological features and active ion channel properties were suggested to contribute to this astonishing processing speed. Here we report an exhaustive study of ultrafast population coding for varying axon initial segment (AIS) location, soma size, and axonal current properties. In particular, we studied their impact on two experimentally observed features 1) precise action potential timing, manifested in a wide-bandwidth dynamic gain, and 2) high-frequency boost under slowly fluctuating correlated input. While the density of axonal channels and their distance from the soma had a very small impact on bandwidth, it could be moderately improved by increasing soma size. When the voltage sensitivity of axonal currents was increased we observed ultrafast coding and high-frequency boost. We conclude that these computationally relevant features are strongly dependent on axonal ion channels’ voltage sensitivity, but not their number or exact location. We point out that ion channel properties, unlike dendrite size, can undergo rapid physiological modification, suggesting that the temporal accuracy of neuronal population encoding could be dynamically regulated. Our results are in line with recent experimental findings in AIS pathologies and establish a framework to study structure-function relations in AIS molecular design.

In large nervous systems, a signal often diverges to hundreds or thousands of neurons. This population’s spike rate can track changes in this common input for frequencies up to several hundred Hertz. This ultrafast population response is experimentally well established and critically impacts cortical information processing. Its underlying biophysical determinants, however, are not understood. Experiments suggest that the ion channels at the axon initial segment strongly contribute to the ultrafast response, but recent theoretical studies emphasize the importance of neuron morphology and the resulting resistive coupling between axon and somato-dendritic compartments. We provide an exhaustive analysis of the population response of a simplified multi-compartment model. We vary the axo-somatic interaction and also active axonal properties and compare models at equivalent working points, avoiding bias. This approach provides a guideline for future experimental and theoretical studies. In this framework, the population response is closely associated with the AP generation speed at the AP initiation site, which is mostly determined by axonal ion channel voltage sensitivity. The resistive axo-somatic coupling has an additional modulatory influence. These insights are expected to hold for encoding mechanisms of more sophisticated models, suggesting that physiological changes to axonal ion channels could modulate the population response rapidly.

Firing rate homeostasis counteracts changes in stability of recurrent neural networks caused by synapse loss in Alzheimer's disease

The impairment of cognitive function in Alzheimer's disease is clearly correlated to synapse loss. However, the mechanisms underlying this correlation are only poorly understood. Here, we investigate how the loss of excitatory synapses in sparsely connected random networks of spiking excitatory and inhibitory neurons alters their dynamical characteristics. Beyond the effects on the activity statistics, we find that the loss of excitatory synapses on excitatory neurons reduces the network's sensitivity to small perturbations. This decrease in sensitivity can be considered as an indication of a reduction of computational capacity. A full recovery of the network's dynamical characteristics and sensitivity can be achieved by firing rate homeostasis, here implemented by an up-scaling of the remaining excitatory-excitatory synapses. Mean-field analysis reveals that the stability of the linearised network dynamics is, in good approximation, uniquely determined by the firing rate, and thereby explains why firing rate homeostasis preserves not only the firing rate but also the network's sensitivity to small perturbations.

The location of the axon initial segment affects the bandwidth of spike initiation dynamics

The dynamics and the sharp onset of action potential (AP) generation have recently been the subject of intense experimental and theoretical investigations. According to the resistive coupling theory, an electrotonic interplay between the site of AP initiation in the axon and the somato-dendritic load determines the AP waveform. This phenomenon not only alters the shape of APs recorded at the soma, but also determines the dynamics of excitability across a variety of time scales. Supporting this statement, here we generalize a previous numerical study and extend it to the quantification of the input-output gain of the neuronal dynamical response. We consider three classes of multicompartmental mathematical models, ranging from ball-and-stick simplified descriptions of neuronal excitability to 3D-reconstructed biophysical models of excitatory neurons of rodent and human cortical tissue. For each model, we demonstrate that increasing the distance between the axonal site of AP initiation and the soma markedly increases the bandwidth of neuronal response properties. We finally consider the Liquid State Machine paradigm, exploring the impact of altering the site of AP initiation at the level of a neuronal population, and demonstrate that an optimal distance exists to boost the computational performance of the network in a simple classification task.

The neurons in the brain encode information through electrical impulses. The performance of a cell in terms of its ability to process and transfer information downstream thus depends heavily on the machinery of initiation of these impulses. In this work, we consider both the cell morphology and the biophysical properties of impulse initiation as the primary parameters that influence information processing in single neurons, as well as in networks. We specifically analyze the location of nerve impulse initiation along the cell’s axon and the way the neuron transfers incoming information. By using single-cell models of various complexity as well as network models, we conclude that information processing is sensitive to the geometrical details of impulse initiation.

Receiver operating characteristic curves for a simple stochastic process that carries a static signal

The detection of a weak signal in the presence of noise is an important problem that is often studied in terms of the receiver operating characteristic (ROC) curve, in which the probability of correct detection is plotted against the probability for a false positive. This kind of analysis is typically applied to the situation in which signal and noise are stochastic variables; the detection problem emerges, however, also often in a context in which both signal and noise are stochastic processes and the (correct or false) detection is said to take place when the process crosses a threshold in a given time window. Here we consider the problem for a combination of a static signal which has to be detected against a dynamic noise process, the well-known Ornstein-Uhlenbeck process. We give exact (but difficult to evaluate) quadrature expressions for the detection rates for false positives and correct detections, investigate systematically a simple sampling approximation suggested earlier, compare to an approximation by Stratonovich for the limit of high threshold, and briefly explore the case of multiplicative signal; all theoretical results are compared to extensive numerical simulations of the corresponding Langevin equation. Our results demonstrate that the sampling approximation provides a reasonable description of the ROC curve for this system, and it clarifies limit cases for the ROC curve.

Homogeneous and Narrow Bandwidth of Spike Initiation in Rat L1 Cortical Interneurons

The cortical layer 1 (L1) contains a population of GABAergic interneurons, considered a key component of information integration, processing, and relaying in neocortical networks. In fact, L1 interneurons combine top–down information with feed-forward sensory inputs in layer 2/3 and 5 pyramidal cells (PCs), while filtering their incoming signals. Despite the importance of L1 for network emerging phenomena, little is known on the dynamics of the spike initiation and the encoding properties of its neurons. Using acute brain tissue slices from the rat neocortex, combined with the analysis of an existing database of model neurons, we investigated the dynamical transfer properties of these cells by sampling an entire population of known “electrical classes” and comparing experiments and model predictions. We found the bandwidth of spike initiation to be significantly narrower than in L2/3 and 5 PCs, with values below 100 cycle/s, but without significant heterogeneity in the cell response properties across distinct electrical types. The upper limit of the neuronal bandwidth was significantly correlated to the mean firing rate, as anticipated from theoretical studies but not reported for PCs. At high spectral frequencies, the magnitude of the neuronal response attenuated as a power-law, with an exponent significantly smaller than what was reported for pyramidal neurons and reminiscent of the dynamics of a “leaky” integrate-and-fire model of spike initiation. Finally, most of our in vitro results matched quantitatively the numerical simulations of the models as a further contribution to independently validate the models against novel experimental data.

Dynamic Gain Analysis Reveals Encoding Deficiencies in Cortical Neurons That Recover from Hypoxia-Induced Spreading Depolarizations

Cortical regions that are damaged by insults, such as ischemia, hypoxia, and trauma, frequently generate spreading depolarization (SD). At the neuronal level, SDs entail complete breakdown of ionic gradients, persisting for seconds to minutes. It is unclear whether these transient events have a more lasting influence on neuronal function. Here, we describe electrophysiological changes in cortical neurons after recovery from hypoxia-induced SD. When examined with standard measures of neuronal excitability several hours after recovery from SD, layer 5 pyramidal neurons in brain slices from mice of either sex appear surprisingly normal. However, we here introduce an additional parameter, dynamic gain, which characterizes the bandwidth of action potential encoding by a neuron, and thereby reflects its potential efficiency in a multineuronal circuit. We find that the ability of neurons that recover from SD to track high-frequency inputs is markedly curtailed; exposure to hypoxia did not have this effect when SD was prevented pharmacologically. Staining for Ankyrin G revealed at least a fourfold decrease in the number of intact axon initial segments in post-SD slices. Since this effect, along with the effect on encoding, was blocked by an inhibitor of the Ca²⁺-dependent enzyme, calpain, we conclude that both effects were mediated by the SD-induced rise in intracellular Ca²⁺ Although effects of calpain activation were detected in the axon initial segment, changes in soma-dendritic compartments may also be involved. Whatever the precise molecular mechanism, our findings indicate that in the context of cortical circuit function, effectiveness of neurons that survive SD may be limited.SIGNIFICANCE STATEMENT Spreading depolarization, which commonly accompanies cortical injury, entails transient massive breakdown of neuronal ionic gradients. The function of cortical neurons that recover from hypoxia-induced spreading depolarization is not obviously abnormal when tested for usual measures of neuronal excitability. However, we now demonstrate that they have a reduced bandwidth, reflecting a significant impairment of their ability to precisely encode high-frequency components of their synaptic input in output spike trains. Thus, neurons that recover from spreading depolarizations are less able to function normally as elements in the multineuronal cortical circuitry. These changes are correlated with activation of the calcium-dependent enzyme, calpain.

Robust closed-loop control of spike-and-wave discharges in a thalamocortical computational model of absence epilepsy

In this paper, we investigate the abatement of spike-and-wave discharges in a thalamocortical model using a closed-loop brain stimulation method. We first explore the complex states and various transitions in the thalamocortical computational model of absence epilepsy by using bifurcation analysis. We demonstrate that the Hopf and double cycle bifurcations are the key dynamical mechanisms of the experimental observed bidirectional communications during absence seizures through top-down cortical excitation and thalamic feedforward inhibition. Then, we formulate the abatement of epileptic seizures to a closed-loop tracking control problem. Finally, we propose a neural network based sliding mode feedback control system to drive the dynamics of pathological cortical area to track the desired normal background activities. The control system is robust to uncertainties and disturbances, and its stability is guaranteed by Lyapunov stability theorem. Our results suggest that the seizure abatement can be modeled as a tracking control problem and solved by a robust closed-loop control method, which provides a promising brain stimulation strategy.

Information transmission of mean and variance coding in integrate-and-fire neurons

Neurons process information by translating continuous signals into patterns of discrete spike times. An open question is how much information these spike times contain about signals which modulate either the mean or the variance of the somatic currents in neurons, as is observed experimentally. Here we calculate the exact information contained in discrete spike times about a continuous signal in both encoding strategies. We show that the information content about mean modulating signals is generally substantially larger than about variance modulating signals for biological parameters. Our analysis further reveals that higher information transmission is associated with a larger proportion of nonlinear signal encoding. Our study measures the complete information content of mean and variance coding and provides a method to determine what fraction of the total information is linearly decodable.

An axon initial segment is required for temporal precision in action potential encoding by neuronal populations

Central neurons initiate action potentials (APs) in the axon initial segment (AIS), a compartment characterized by a high concentration of voltage-dependent ion channels and specialized cytoskeletal anchoring proteins arranged in a regular nanoscale pattern. Although the AIS was a key evolutionary innovation in neurons, the functional benefits it confers are not clear. Using a mutation of the AIS cytoskeletal protein βIV-spectrin, we here establish an in vitro model of neurons with a perturbed AIS architecture that retains nanoscale order but loses the ability to maintain a high Na_V density. Combining experiments and simulations, we show that a high Na_V density in the AIS is not required for axonal AP initiation; it is, however, crucial for a high bandwidth of information encoding and AP timing precision. Our results provide the first experimental demonstration of axonal AP initiation without high axonal channel density and suggest that increasing the bandwidth of the neuronal code and, hence, the computational efficiency of network function, was a major benefit of the evolution of the AIS.

Micropools of reliable area MT neurons explain rapid motion detection

Many models of perceptually based decisions postulate that actions are initiated when accumulated sensory signals reach a threshold level of activity. These models have received considerable neurophysiological support from recordings of individual neurons while animals are engaged in motion discrimination tasks. These experiments have found that the activity of neurons in a particular visual area strongly associated with motion processing (MT), when pooled over hundreds of milliseconds, is sufficient to explain behavioral timing and performance. However, this level of pooling may be problematic for urgent perceptual decisions in which rapid detection dictates temporally precise integration. In this paper, we explore the physiological basis of one such task in which macaques detected brief (~70 ms) transients of coherent motion within ~240 ms. We find that a simple linear summation model based on realistic stimulus responses of as few as 40 correlated neurons can predict the reliability and timing of rapid motion detection. The model naturally reproduces a distinctive physiological relationship observed in rapid detection tasks in which the individual neurons with the most reliable stimulus responses are also the most predictive of impending behavioral choices. Remarkably, we observed this relationship across our simulated neuronal populations even when all neurons within the pool were weighted equally with respect to readout. These results demonstrate that small numbers of reliable sensory neurons can dominate perceptual judgments without any explicit reliability based weighting and are sufficient to explain the accuracy, latency, and temporal precision of rapid detection. NEW & NOTEWORTHY Computational and psychophysical models suggest that performance in many perceptual tasks may be based on the preferential sampling of reliable neurons. Recent studies of MT neurons during rapid motion detection, in which only those neurons with the most reliable sensory responses were strongly predictive of the animals' decisions, seemingly support this notion. Here we show that a simple threshold model without explicit reliability biases can explain both the behavioral accuracy and precision of these detections and the distribution of sensory- and choice-related signals across neurons.

Dynamical response properties of neocortical neurons to conductance-driven time-varying inputs

Ensembles of cortical neurons can track fast-varying inputs and relay them in their spike trains, far beyond the cut-off imposed by membrane passive electrical properties and mean firing rates. Initially explored in silico and later demonstrated experimentally, investigating how neurons respond to sinusoidally modulated stimuli provides a deeper insight into spike initiation mechanisms and information processing than conventional F-I curve methodologies. Besides net membrane currents, physiological synaptic inputs can also induce a stimulus-dependent modulation of the total membrane conductance, which is not reproduced by standard current-clamp protocols. Here, we investigated whether rat cortical neurons can track fast temporal modulations over a noisy conductance background. We also determined input-output transfer properties over a range of conditions, including: distinct presynaptic activation rates, postsynaptic firing rates and variability and type of temporal modulations. We found a very broad signal transfer bandwidth across all conditions, similar large cut-off frequencies and power-law attenuations of fast-varying inputs. At slow and intermediate input modulations, the response gain decreased for increasing output mean firing rates. The gain also decreased significantly for increasing intensities of background synaptic activity, thus generalising earlier studies on F-I curves. We also found a direct correlation between the action potentials' onset rapidness and the neuronal bandwidth. Our novel results extend previous investigations of dynamical response properties to non-stationary and conductance-driven conditions, and provide computational neuroscientists with a novel set of observations that models must capture when aiming to replicate cortical cellular excitability.

Noisy Juxtacellular Stimulation In Vivo Leads to Reliable Spiking and Reveals High-Frequency Coding in Single Neurons

Single cells in the motor and somatosensory cortex of rats were stimulated in vivo with broadband fluctuating currents applied juxtacellularly. Unlike the DC current steps used previously, fluctuating stimulation currents reliably evoked spike trains with precise timing of individual spikes. Fluctuating currents resulted in strong cellular responses at stimulation frequencies beyond the inverse membrane time constant and the mean firing rate of the neuron. Neuronal firing was associated with high rates of information transmission, even for the high-frequency components of the stimulus. Such response characteristics were also revealed in additional experiments with sinusoidal juxtacellular stimulation. For selected cells, we could reproduce these statistics with compartmental models of varying complexity. We also developed a method to generate Gaussian stimuli that evoke spike trains with prescribed spike times (under the constraint of a certain rate and coefficient of variation) and exemplify its ability to achieve precise and reliable spiking in cortical neurons in vivo Our results demonstrate a novel method for precise control of spike timing by juxtacellular stimulation, confirm and extend earlier conclusions from ex vivo work about the capacity of cortical neurons to generate precise discharges, and contribute to the understanding of the biophysics of information transfer of single neurons in vivo at high frequencies.

SIGNIFICANCE STATEMENT

Nanostimulation of single identified neurons in vivo can control spike frequency parametrically and, surprisingly, can even bias the animal's behavioral response. Here, we extend this stimulation protocol to time-dependent broadband noise stimulation in sensory and motor cortices of rat. In response to such stimuli, we found increased temporal spike-time reliability. The information transmission properties reveal, both experimentally and theoretically, that the neurons support high-frequency stimulation beyond the inverse membrane time. Generating a stimulus using the neuron's response properties, we could evoke prescribed spike times with high precision. Our work helps to establish a novel method for precise temporal control of single-cell spiking and provides a simplified biophysical description of single-neuron spiking under time-dependent in vivo-like stimulation.

Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise

A neuron receives input from other neurons via electrical pulses, so-called spikes. The pulse-like nature of the input is frequently neglected in analytical studies; instead, the input is usually approximated to be Gaussian. Recent experimental studies have shown, however, that an assumption underlying this approximation is often not met: Individual presynaptic spikes can have a significant effect on a neuron's dynamics. It is thus desirable to explicitly account for the pulse-like nature of neural input, i.e. consider neurons driven by a shot noise - a long-standing problem that is mathematically challenging. In this work, we exploit the fact that excitatory shot noise with exponentially distributed weights can be obtained as a limit case of dichotomous noise, a Markovian two-state process. This allows us to obtain novel exact expressions for the stationary voltage density and the moments of the interspike-interval density of general integrate-and-fire neurons driven by such an input. For the special case of leaky integrate-and-fire neurons, we also give expressions for the power spectrum and the linear response to a signal. We verify and illustrate our expressions by comparison to simulations of leaky-, quadratic- and exponential integrate-and-fire neurons.

Encoding of High Frequencies Improves with Maturation of Action Potential Generation in Cultured Neocortical Neurons

The ability of neocortical neurons to detect and encode rapid changes at their inputs is crucial for basic neuronal computations, such as coincidence detection, precise synchronization of activity and spike-timing dependent plasticity. Indeed, populations of cortical neurons can respond to subtle changes of the input very fast, on a millisecond time scale. Theoretical studies and model simulations linked the encoding abilities of neuronal populations to the fast onset dynamics of action potentials (APs). Experimental results support this idea, however mechanisms of fast onset of APs in cortical neurons remain elusive. Studies in neuronal cultures, that are allowing for accurate control over conditions of growth and microenvironment during the development of neurons and provide better access to the spike initiation zone, may help to shed light on mechanisms of AP generation and encoding. Here we characterize properties of AP encoding in neocortical neurons grown for 11-25 days in culture. We show that encoding of high frequencies improves upon culture maturation, which is accompanied by the development of passive electrophysiological properties and AP generation. The onset of APs becomes faster with culture maturation. Statistical analysis using correlations and linear model approaches identified the onset dynamics of APs as a major predictor of age-dependent changes of encoding. Encoding of high frequencies strongly correlated also with the input resistance of neurons. Finally, we show that maturation of encoding properties of neurons in cultures is similar to the maturation of encoding in neurons studied in slices. These results show that maturation of AP generators and encoding is, to a large extent, determined genetically and takes place even without normal micro-environment and activity of the whole brain in vivo. This establishes neuronal cultures as a valid experimental model for studying mechanisms of AP generation and encoding, and their maturation.

From Neuron Biophysics to Orientation Selectivity in Electrically Coupled Networks of Neocortical L2/3 Large Basket Cells

In the neocortex, inhibitory interneurons of the same subtype are electrically coupled with each other via dendritic gap junctions (GJs). The impact of multiple GJs on the biophysical properties of interneurons and thus on their input processing is unclear. The present experimentally based theoretical study examined GJs in L2/3 large basket cells (L2/3 LBCs) with 3 goals in mind: (1) To evaluate the errors due to GJs in estimating the cable properties of individual L2/3 LBCs and suggest ways to correct these errors when modeling these cells and the networks they form; (2) to bracket the GJ conductance value (0.05-0.25 nS) and membrane resistivity (10 000-40 000 Ω cm(2)) of L2/3 LBCs; these estimates are tightly constrained by in vitro input resistance (131 ± 18.5 MΩ) and the coupling coefficient (1-3.5%) of these cells; and (3) to explore the functional implications of GJs, and show that GJs: (i) dynamically modulate the effective time window for synaptic integration; (ii) improve the axon's capability to encode rapid changes in synaptic inputs; and (iii) reduce the orientation selectivity, linearity index, and phase difference of L2/3 LBCs. Our study provides new insights into the role of GJs and calls for caution when using in vitro measurements for modeling electrically coupled neuronal networks.

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.

Impact of membrane bistability on dynamical response of neuronal populations

Neurons in many brain areas can develop a pronounced depolarized state of membrane potential (up state) in addition to the normal hyperpolarized state near the resting potential. The influence of the up state on signal encoding, however, is not well investigated. Here we construct a one-dimensional bistable neuron model and calculate the linear dynamical response to noisy oscillatory inputs analytically. We find that with the appearance of an up state, the transmission function is enhanced by the emergence of a local maximum at some optimal frequency and the phase lag relative to the input signal is reduced. We characterize the dependence of the enhancement of frequency response on intrinsic dynamics and on the occupancy of the up state.

Neuronal morphology generates high-frequency firing resonance

The attenuation of neuronal voltage responses to high-frequency current inputs by the membrane capacitance is believed to limit single-cell bandwidth. However, neuronal populations subject to stochastic fluctuations can follow inputs beyond this limit. We investigated this apparent paradox theoretically and experimentally using Purkinje cells in the cerebellum, a motor structure that benefits from rapid information transfer. We analyzed the modulation of firing in response to the somatic injection of sinusoidal currents. Computational modeling suggested that, instead of decreasing with frequency, modulation amplitude can increase up to high frequencies because of cellular morphology. Electrophysiological measurements in adult rat slices confirmed this prediction and displayed a marked resonance at 200 Hz. We elucidated the underlying mechanism, showing that the two-compartment morphology of the Purkinje cell, interacting with a simple spiking mechanism and dendritic fluctuations, is sufficient to create high-frequency signal amplification. This mechanism, which we term morphology-induced resonance, is selective for somatic inputs, which in the Purkinje cell are exclusively inhibitory. The resonance sensitizes Purkinje cells in the frequency range of population oscillations observed in vivo.

Action potential initiation in a multi-compartmental model with cooperatively gating Na channels in the axon initial segment

Somatic action potentials (AP) of cortical pyramidal neurons have characteristically high onset-rapidness. The onset of the AP waveform is an indirect measure for the ability of a neuron to respond to temporally fast-changing stimuli. Theoretical studies on the pyramidal neuron response usually involves a canonical Hodgkin-Huxley (HH) type ion channel gating model, which assumes statistically independent gating of each individual channel. However, cooperative activity of ion channels are observed for various cell types, meaning that the activity (e.g. opening) of one channel triggers the activity (e.g. opening) of a certain fraction of its neighbors and hence, these groups of channels behave as a unit. In this study, we describe a multi-compartmental conductance-based model with cooperatively gating voltage-gated Na channels in the axon initial segment. Our model successfully reproduced the somatic sharp AP onsets of cortical pyramidal neurons. The onset latencies from the initiation site to the soma and the conduction velocities were also in agreement with the previous experimental studies.

Advantages and limitations of the use of optogenetic approach in studying fast-scale spike encoding

Understanding single-neuron computations and encoding performed by spike-generation mechanisms of cortical neurons is one of the central challenges for cell electrophysiology and computational neuroscience. An established paradigm to study spike encoding in controlled conditions in vitro uses intracellular injection of a mixture of signals with fluctuating currents that mimic in vivo-like background activity. However this technique has two serious limitations: it uses current injection, while synaptic activation leads to changes of conductance, and current injection is technically most feasible in the soma, while the vast majority of synaptic inputs are located on the dendrites. Recent progress in optogenetics provides an opportunity to circumvent these limitations. Transgenic expression of light-activated ionic channels, such as Channelrhodopsin2 (ChR2), allows induction of controlled conductance changes even in thin distant dendrites. Here we show that photostimulation provides a useful extension of the tools to study neuronal encoding, but it has its own limitations. Optically induced fluctuating currents have a low cutoff (~70Hz), thus limiting the dynamic range of frequency response of cortical neurons. This leads to severe underestimation of the ability of neurons to phase-lock their firing to high frequency components of the input. This limitation could be worked around by using short (2 ms) light stimuli which produce membrane potential responses resembling EPSPs by their fast onset and prolonged decay kinetics. We show that combining application of short light stimuli to different parts of dendritic tree for mimicking distant EPSCs with somatic injection of fluctuating current that mimics fluctuations of membrane potential in vivo, allowed us to study fast encoding of artificial EPSPs photoinduced at different distances from the soma. We conclude that dendritic photostimulation of ChR2 with short light pulses provides a powerful tool to investigate population encoding of simulated synaptic potentials generated in dendrites at different distances from the soma.

Cortical Specializations Underlying Fast Computations

The time course of behaviorally relevant environmental events sets temporal constraints on neuronal processing. How does the mammalian brain make use of the increasingly complex networks of the neocortex, while making decisions and executing behavioral reactions within a reasonable time? The key parameter determining the speed of computations in neuronal networks is a time interval that neuronal ensembles need to process changes at their input and communicate results of this processing to downstream neurons. Theoretical analysis identified basic requirements for fast processing: use of neuronal populations for encoding, background activity, and fast onset dynamics of action potentials in neurons. Experimental evidence shows that populations of neocortical neurons fulfil these requirements. Indeed, they can change firing rate in response to input perturbations very quickly, within 1 to 3 ms, and encode high-frequency components of the input by phase-locking their spiking to frequencies up to 300 to 1000 Hz. This implies that time unit of computations by cortical ensembles is only few, 1 to 3 ms, which is considerably faster than the membrane time constant of individual neurons. The ability of cortical neuronal ensembles to communicate on a millisecond time scale allows for complex, multiple-step processing and precise coordination of neuronal activity in parallel processing streams, while keeping the speed of behavioral reactions within environmentally set temporal constraints.

A Markov model for the temporal dynamics of balanced random networks of finite size

The balanced state of recurrent networks of excitatory and inhibitory spiking neurons is characterized by fluctuations of population activity about an attractive fixed point. Numerical simulations show that these dynamics are essentially nonlinear, and the intrinsic noise (self-generated fluctuations) in networks of finite size is state-dependent. Therefore, stochastic differential equations with additive noise of fixed amplitude cannot provide an adequate description of the stochastic dynamics. The noise model should, rather, result from a self-consistent description of the network dynamics. Here, we consider a two-state Markovian neuron model, where spikes correspond to transitions from the active state to the refractory state. Excitatory and inhibitory input to this neuron affects the transition rates between the two states. The corresponding nonlinear dependencies can be identified directly from numerical simulations of networks of leaky integrate-and-fire neurons, discretized at a time resolution in the sub-millisecond range. Deterministic mean-field equations, and a noise component that depends on the dynamic state of the network, are obtained from this model. The resulting stochastic model reflects the behavior observed in numerical simulations quite well, irrespective of the size of the network. In particular, a strong temporal correlation between the two populations, a hallmark of the balanced state in random recurrent networks, are well represented by our model. Numerical simulations of such networks show that a log-normal distribution of short-term spike counts is a property of balanced random networks with fixed in-degree that has not been considered before, and our model shares this statistical property. Furthermore, the reconstruction of the flow from simulated time series suggests that the mean-field dynamics of finite-size networks are essentially of Wilson-Cowan type. We expect that this novel nonlinear stochastic model of the interaction between neuronal populations also opens new doors to analyze the joint dynamics of multiple interacting networks.

Injection of fully-defined signal mixtures: a novel high-throughput tool to study neuronal encoding and computations

Understanding of how neurons transform fluctuations of membrane potential, reflecting input activity, into spike responses, which communicate the ultimate results of single-neuron computation, is one of the central challenges for cellular and computational neuroscience. To study this transformation under controlled conditions, previous work has used a signal immersed in noise paradigm where neurons are injected with a current consisting of fluctuating noise that mimics on-going synaptic activity and a systematic signal whose transmission is studied. One limitation of this established paradigm is that it is designed to examine the encoding of only one signal under a specific, repeated condition. As a result, characterizing how encoding depends on neuronal properties, signal parameters, and the interaction of multiple inputs is cumbersome. Here we introduce a novel fully-defined signal mixture paradigm, which allows us to overcome these problems. In this paradigm, current for injection is synthetized as a sum of artificial postsynaptic currents (PSCs) resulting from the activity of a large population of model presynaptic neurons. PSCs from any presynaptic neuron(s) can be now considered as "signal", while the sum of all other inputs is considered as "noise". This allows us to study the encoding of a large number of different signals in a single experiment, thus dramatically increasing the throughput of data acquisition. Using this novel paradigm, we characterize the detection of excitatory and inhibitory PSCs from neuronal spike responses over a wide range of amplitudes and firing-rates. We show, that for moderately-sized neuronal populations the detectability of individual inputs is higher for excitatory than for inhibitory inputs during the 2-5 ms following PSC onset, but becomes comparable after 7-8 ms. This transient imbalance of sensitivity in favor of excitation may enhance propagation of balanced signals through neuronal networks. Finally, we discuss several open questions that this novel high-throughput paradigm may address.

Dendrites impact the encoding capabilities of the axon

This study highlights a new and powerful direct impact of the dendritic tree (the input region of neurons) on the encoding capability of the axon (the output region). We show that the size of the dendritic arbors (its impedance load) strongly modulates the shape of the action potential (AP) onset at the axon initial segment; it is accelerated in neurons with larger dendritic surface area. AP onset rapidness is key in determining the capability of the axonal spikes to encode (phase lock to) rapid changes in synaptic inputs. Hence, our findings imply that neurons with larger dendritic arbors have improved encoding capabilities. This "dendritic size effect" was explored both analytically as well as numerically, in simplified and detailed models of 3D reconstructed layer 2/3 cortical pyramidal cells of rats and humans. The cutoff frequency of spikes phase locking to modulated inputs increased from 100 to 200 Hz in pyramidal cells of young rats to 400-600 Hz in human cells. In the latter case, phase locking reached close to 1 KHz in in vivo-like conditions. This work highlights new and functionally profound cross talk between the dendritic tree and the axon initial segment, providing new understanding of neurons as sophisticated nonlinear input/output devices.

Static frequency tuning accounts for changes in neural synchrony evoked by transient communication signals

Although communication signals often vary continuously on the underlying signal parameter, they are perceived as distinct categories. We here report the opposite case where an electrocommunication signal is encoded in four distinct regimes, although the behavior described to date does not show distinct categories. In particular, we studied the encoding of chirps by P-unit afferents in the weakly electric fish Apteronotus leptorhynchus. These fish generate an electric organ discharge that oscillates at a certain individual-specific frequency. The interaction of two fish in communication contexts leads to the emergence of a beating amplitude modulation (AM) at the frequency difference between the two individual signals. This frequency difference represents the social context of the encounter. Chirps are transient increases of the fish's frequency leading to transient changes in the frequency of the AM. We stimulated the cells with the same chirp on different, naturally occurring backgrounds beats. The P-units responded either by synchronization or desynchronization depending on the background. Although the duration of a chirp is often shorter than a full cycle of the AM it elicits, the distinct responses of the P-units to the chirp can be predicted solely from the frequency of the AM based on the static frequency tuning of the cells.

Dynamical models of cortical circuits

Cortical neurons operate within recurrent neuronal circuits. Dissecting their operation is key to understanding information processing in the cortex and requires transparent and adequate dynamical models of circuit function. Convergent evidence from experimental and theoretical studies indicates that strong feedback inhibition shapes the operating regime of cortical circuits. For circuits operating in inhibition-dominated regimes, mathematical and computational studies over the past several years achieved substantial advances in understanding response modulation and heterogeneity, emergent stimulus selectivity, inter-neuron correlations, and microstate dynamics. The latter indicate a surprisingly strong dependence of the collective circuit dynamics on the features of single neuron action potential generation. New approaches are needed to definitely characterize the cortical operating regime.

Single neuron dynamics and computation

At the single neuron level, information processing involves the transformation of input spike trains into an appropriate output spike train. Building upon the classical view of a neuron as a threshold device, models have been developed in recent years that take into account the diverse electrophysiological make-up of neurons and accurately describe their input-output relations. Here, we review these recent advances and survey the computational roles that they have uncovered for various electrophysiological properties, for dendritic arbor anatomy as well as for short-term synaptic plasticity.

How pattern formation in ring networks of excitatory and inhibitory spiking neurons depends on the input current regime

Pattern formation, i.e., the generation of an inhomogeneous spatial activity distribution in a dynamical system with translation invariant structure, is a well-studied phenomenon in neuronal network dynamics, specifically in neural field models. These are population models to describe the spatio-temporal dynamics of large groups of neurons in terms of macroscopic variables such as population firing rates. Though neural field models are often deduced from and equipped with biophysically meaningful properties, a direct mapping to simulations of individual spiking neuron populations is rarely considered. Neurons have a distinct identity defined by their action on their postsynaptic targets. In its simplest form they act either excitatorily or inhibitorily. When the distribution of neuron identities is assumed to be periodic, pattern formation can be observed, given the coupling strength is supracritical, i.e., larger than a critical weight. We find that this critical weight is strongly dependent on the characteristics of the neuronal input, i.e., depends on whether neurons are mean- or fluctuation driven, and different limits in linearizing the full non-linear system apply in order to assess stability. In particular, if neurons are mean-driven, the linearization has a very simple form and becomes independent of both the fixed point firing rate and the variance of the input current, while in the very strongly fluctuation-driven regime the fixed point rate, as well as the input mean and variance are important parameters in the determination of the critical weight. We demonstrate that interestingly even in "intermediate" regimes, when the system is technically fluctuation-driven, the simple linearization neglecting the variance of the input can yield the better prediction of the critical coupling strength. We moreover analyze the effects of structural randomness by rewiring individual synapses or redistributing weights, as well as coarse-graining on the formation of inhomogeneous activity patterns.

Fast computations in cortical ensembles require rapid initiation of action potentials

The abilities of neuronal populations to encode rapidly varying stimuli and respond quickly to abrupt input changes are crucial for basic neuronal computations, such as coincidence detection, grouping by synchrony, and spike-timing-dependent plasticity, as well as for the processing speed of neuronal networks. Theoretical analyses have linked these abilities to the fast-onset dynamics of action potentials (APs). Using a combination of whole-cell recordings from rat neocortical neurons and computer simulations, we provide the first experimental evidence for this conjecture and prove its validity for the case of distal AP initiation in the axon initial segment (AIS), typical for cortical neurons. Neocortical neurons with fast-onset APs in the soma can phase-lock their population firing to signal frequencies up to ∼300-400 Hz and respond within 1-2 ms to subtle changes of input current. The ability to encode high frequencies and response speed were dramatically reduced when AP onset was slowed by experimental manipulations or was intrinsically slow due to immature AP generation mechanisms. Multicompartment conductance-based models reproducing the initiation of spikes in the AIS could encode high frequencies only if AP onset was fast at the initiation site (e.g., attributable to cooperative gating of a fraction of sodium channels) but not when fast onset of somatic AP was produced solely by backpropagation. We conclude that fast-onset dynamics is a genuine property of cortical AP generators. It enables fast computations in cortical circuits that are rich in recurrent connections both within each region and across the hierarchy of areas.

Oscillatory threshold logic

In the 1940s, the first generation of modern computers used vacuum tube oscillators as their principle components, however, with the development of the transistor, such oscillator based computers quickly became obsolete. As the demand for faster and lower power computers continues, transistors are themselves approaching their theoretical limit and emerging technologies must eventually supersede them. With the development of optical oscillators and Josephson junction technology, we are again presented with the possibility of using oscillators as the basic components of computers, and it is possible that the next generation of computers will be composed almost entirely of oscillatory devices. Here, we demonstrate how coupled threshold oscillators may be used to perform binary logic in a manner entirely consistent with modern computer architectures. We describe a variety of computational circuitry and demonstrate working oscillator models of both computation and memory.

Decorrelation of neural-network activity by inhibitory feedback

Correlations in spike-train ensembles can seriously impair the encoding of information by their spatio-temporal structure. An inevitable source of correlation in finite neural networks is common presynaptic input to pairs of neurons. Recent studies demonstrate that spike correlations in recurrent neural networks are considerably smaller than expected based on the amount of shared presynaptic input. Here, we explain this observation by means of a linear network model and simulations of networks of leaky integrate-and-fire neurons. We show that inhibitory feedback efficiently suppresses pairwise correlations and, hence, population-rate fluctuations, thereby assigning inhibitory neurons the new role of active decorrelation. We quantify this decorrelation by comparing the responses of the intact recurrent network (feedback system) and systems where the statistics of the feedback channel is perturbed (feedforward system). Manipulations of the feedback statistics can lead to a significant increase in the power and coherence of the population response. In particular, neglecting correlations within the ensemble of feedback channels or between the external stimulus and the feedback amplifies population-rate fluctuations by orders of magnitude. The fluctuation suppression in homogeneous inhibitory networks is explained by a negative feedback loop in the one-dimensional dynamics of the compound activity. Similarly, a change of coordinates exposes an effective negative feedback loop in the compound dynamics of stable excitatory-inhibitory networks. The suppression of input correlations in finite networks is explained by the population averaged correlations in the linear network model: In purely inhibitory networks, shared-input correlations are canceled by negative spike-train correlations. In excitatory-inhibitory networks, spike-train correlations are typically positive. Here, the suppression of input correlations is not a result of the mere existence of correlations between excitatory (E) and inhibitory (I) neurons, but a consequence of a particular structure of correlations among the three possible pairings (EE, EI, II).

A small fraction of strongly cooperative sodium channels boosts neuronal encoding of high frequencies

Generation of action potentials (APs) is a crucial step in neuronal information processing. Existing biophysical models for AP generation almost universally assume that individual voltage-gated sodium channels operate statistically independently, and their avalanche-like opening that underlies AP generation is coordinated only through the transmembrane potential. However, biological ion channels of various types can exhibit strongly cooperative gating when clustered. Cooperative gating of sodium channels has been suggested to explain rapid onset dynamics and large threshold variability of APs in cortical neurons. It remains however unknown whether these characteristic properties of cortical APs can be reproduced if only a fraction of channels express cooperativity, and whether the presence of cooperative channels has an impact on encoding properties of neuronal populations. To address these questions we have constructed a conductance-based neuron model in which we continuously varied the size of a fraction [Formula: see text] of sodium channels expressing cooperativity and the strength of coupling between cooperative channels [Formula: see text]. We show that starting at a critical value of the coupling strength [Formula: see text], the activation curve of sodium channels develops a discontinuity at which opening of all coupled channels becomes an all-or-none event, leading to very rapid AP onsets. Models with a small fraction, [Formula: see text], of strongly cooperative channels generate APs with the most rapid onset dynamics. In this regime APs are triggered by simultaneous opening of the cooperative channel fraction and exhibit a pronounced biphasic waveform often observed in cortical neurons. We further show that presence of a small fraction of cooperative Na+ channels significantly improves the ability of neuronal populations to phase-lock their firing to high frequency input fluctuation. We conclude that presence of a small fraction of strongly coupled sodium channels can explain characteristic features of cortical APs and has a functional impact of enhancing the spike encoding of rapidly varying signals.

Beyond statistical significance: implications of network structure on neuronal activity

It is a common and good practice in experimental sciences to assess the statistical significance of measured outcomes. For this, the probability of obtaining the actual results is estimated under the assumption of an appropriately chosen null-hypothesis. If this probability is smaller than some threshold, the results are deemed statistically significant and the researchers are content in having revealed, within their own experimental domain, a “surprising” anomaly, possibly indicative of a hitherto hidden fragment of the underlying “ground-truth”. What is often neglected, though, is the actual importance of these experimental outcomes for understanding the system under investigation. We illustrate this point by giving practical and intuitive examples from the field of systems neuroscience. Specifically, we use the notion of embeddedness to quantify the impact of a neuron's activity on its downstream neurons in the network. We show that the network response strongly depends on the embeddedness of stimulated neurons and that embeddedness is a key determinant of the importance of neuronal activity on local and downstream processing. We extrapolate these results to other fields in which networks are used as a theoretical framework.

Ultrafast population encoding by cortical neurons

The processing speed of the brain depends on the ability of neurons to rapidly relay input changes. Previous theoretical and experimental studies of the timescale of population firing rate responses arrived at controversial conclusions, some advocating an ultrafast response scale but others arguing for an inherent disadvantage of mean encoded signals for rapid detection of the stimulus onset. Here we assessed the timescale of population firing rate responses of neocortical neurons in experiments performed in the time domain and the frequency domain in vitro and in vivo. We show that populations of neocortical neurons can alter their firing rate within 1 ms in response to somatically delivered weak current signals presented on a fluctuating background. Signals with amplitudes of miniature postsynaptic currents can be robustly and rapidly detected in the population firing. We further show that population firing rate of neurons of rat visual cortex in vitro and cat visual cortex in vivo can reliably encode weak signals varying at frequencies up to ∼200-300 Hz, or ∼50 times faster than the firing rate of individual neurons. These results provide coherent evidence for the ultrafast, millisecond timescale of cortical population responses. Notably, fast responses to weak stimuli are limited to the mean encoding. Rapid detection of current variance changes requires extraordinarily large signal amplitudes. Our study presents conclusive evidence showing that cortical neurons are capable of rapidly relaying subtle mean current signals. This provides a vital mechanism for the propagation of rate-coded information within and across brain areas.

Representation of dynamical stimuli in populations of threshold neurons

Many sensory or cognitive events are associated with dynamic current modulations in cortical neurons. This raises an urgent demand for tractable model approaches addressing the merits and limits of potential encoding strategies. Yet, current theoretical approaches addressing the response to mean- and variance-encoded stimuli rarely provide complete response functions for both modes of encoding in the presence of correlated noise. Here, we investigate the neuronal population response to dynamical modifications of the mean or variance of the synaptic bombardment using an alternative threshold model framework. In the variance and mean channel, we provide explicit expressions for the linear and non-linear frequency response functions in the presence of correlated noise and use them to derive population rate response to step-like stimuli. For mean-encoded signals, we find that the complete response function depends only on the temporal width of the input correlation function, but not on other functional specifics. Furthermore, we show that both mean- and variance-encoded signals can relay high-frequency inputs, and in both schemes step-like changes can be detected instantaneously. Finally, we obtain the pairwise spike correlation function and the spike triggered average from the linear mean-evoked response function. These results provide a maximally tractable limiting case that complements and extends previous results obtained in the integrate and fire framework.

Sensory stimuli in our environment are represented in the brain as input current changes to neurons. For example, a periodic bar pattern in the visual field leads to periodic current modulations in the visual cortex. Therefore, models describing the ability of neurons to represent incoming stimuli can offer important clues about how sensory stimuli are processed by the brain. As anyone who has used an old-fashioned radio can attest, there is not just one but multiple ways to encode a signal, e.g. the familiar AM and FM channels. But what are the potential encoding channels in the cortex? A signal could modify the neuronal input current in two distinct ways: it could act either on the mean or the variance of the current. Using a minimal model framework, which can reproduce many features of neuronal activity, we find that both encoding schemes could be equally potent in transmitting slow and fast signals. This allows us to describe how input signals of any functional form give rise to collective firing rate changes in populations of neurons.

Spike onset dynamics and response speed in neuronal populations

Recent studies of cortical neurons driven by fluctuating currents revealed cutoff frequencies for action potential encoding of several hundred Hz. Theoretical studies of biophysical neuron models have predicted a much lower cutoff frequency of the order of average firing rate or the inverse membrane time constant. The biophysical origin of the observed high cutoff frequencies is thus not well understood. Here we introduce a neuron model with dynamical action potential generation, in which the linear response can be analytically calculated for uncorrelated synaptic noise. We find that the cutoff frequencies increase to very large values when the time scale of action potential initiation becomes short.

The dynamics of integrate-and-fire: mean versus variance modulations and dependence on baseline parameters

The leaky integrate-and-fire (LIF) is the simplest neuron model that captures the essential properties of neuronal signaling. Yet common intuitions are inadequate to explain basic properties of LIF responses to sinusoidal modulations of the input. Here we examine responses to low and moderate frequency modulations of both the mean and variance of the input current and quantify how these responses depend on baseline parameters. Across parameters, responses to modulations in the mean current are low pass, approaching zero in the limit of high frequencies. For very low baseline firing rates, the response cutoff frequency matches that expected from membrane integration. However, the cutoff shows a rapid, supralinear increase with firing rate, with a steeper increase in the case of lower noise. For modulations of the input variance, the gain at high frequency remains finite. Here, we show that the low-frequency responses depend strongly on baseline parameters and derive an analytic condition specifying the parameters at which responses switch from being dominated by low versus high frequencies. Additionally, we show that the resonant responses for variance modulations have properties not expected for common oscillatory resonances: they peak at frequencies higher than the baseline firing rate and persist when oscillatory spiking is disrupted by high noise. Finally, the responses to mean and variance modulations are shown to have a complementary dependence on baseline parameters at higher frequencies, resulting in responses to modulations of Poisson input rates that are independent of baseline input statistics.

From spiking neuron models to linear-nonlinear models

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.

Rate Dynamics of Leaky Integrate-and-Fire Neurons with Strong Synapses

Firing-rate models provide a practical tool for studying the dynamics of trial- or population-averaged neuronal signals. A wealth of theoretical and experimental studies has been dedicated to the derivation or extraction of such models by investigating the firing-rate response characteristics of ensembles of neurons. The majority of these studies assumes that neurons receive input spikes at a high rate through weak synapses (diffusion approximation). For many biological neural systems, however, this assumption cannot be justified. So far, it is unclear how time-varying presynaptic firing rates are transmitted by a population of neurons if the diffusion assumption is dropped. Here, we numerically investigate the stationary and non-stationary firing-rate response properties of leaky integrate-and-fire neurons receiving input spikes through excitatory synapses with alpha-function shaped postsynaptic currents for strong synaptic weights. Input spike trains are modeled by inhomogeneous Poisson point processes with sinusoidal rate. Average rates, modulation amplitudes, and phases of the period-averaged spike responses are measured for a broad range of stimulus, synapse, and neuron parameters. Across wide parameter regions, the resulting transfer functions can be approximated by a linear first-order low-pass filter. Below a critical synaptic weight, the cutoff frequencies are approximately constant and determined by the synaptic time constants. Only for synapses with unrealistically strong weights are the cutoff frequencies significantly increased. To account for stimuli with larger modulation depths, we combine the measured linear transfer function with the nonlinear response characteristics obtained for stationary inputs. The resulting linear-nonlinear model accurately predicts the population response for a variety of non-sinusoidal stimuli.

How connectivity, background activity, and synaptic properties shape the cross-correlation between spike trains

Functional interactions between neurons in vivo are often quantified by cross-correlation functions (CCFs) between their spike trains. It is therefore essential to understand quantitatively how CCFs are shaped by different factors, such as connectivity, synaptic parameters, and background activity. Here, we study the CCF between two neurons using analytical calculations and numerical simulations. We quantify the role of synaptic parameters, such as peak conductance, decay time, and reversal potential, and analyze how various patterns of connectivity influence CCF shapes. In particular, we find that the symmetry of the CCF distinguishes in general, but not always, the case of shared inputs between two neurons from the case in which they are directly synaptically connected. We systematically examine the influence of background synaptic inputs from the surrounding network that set the baseline firing statistics of the neurons and modulate their response properties. We find that variations in the background noise modify the amplitude of the cross-correlation function as strongly as variations of synaptic strength. In particular, we show that the postsynaptic neuron spiking regularity has a pronounced influence on CCF amplitude. This suggests an efficient and flexible mechanism for modulating functional interactions.

Precisely timed signal transmission in neocortical networks with reliable intermediate-range projections

The mammalian neocortex has a remarkable ability to precisely reproduce behavioral sequences or to reliably retrieve stored information. In contrast, spiking activity in behaving animals shows a considerable trial-to-trial variability and temporal irregularity. The signal propagation and processing underlying these conflicting observations is based on fundamental neurophysiological processes like synaptic transmission, signal integration within single cells, and spike formation. Each of these steps in the neuronal signaling chain has been studied separately to a great extend, but it has been difficult to judge how they interact and sum up in active sub-networks of neocortical cells. In the present study, we experimentally assessed the precision and reliability of small neocortical networks consisting of trans-columnar, intermediate-range projections (200-1000 mum) on a millisecond time-scale. Employing photo-uncaging of glutamate in acute slices, we activated a number of distant presynaptic cells in a spatio-temporally precisely controlled manner, while monitoring the resulting membrane potential fluctuations of a postsynaptic cell. We found that signal integration in this part of the network is highly reliable and temporally precise. As numerical simulations showed, the residual membrane potential variability can be attributed to amplitude variability in synaptic transmission and may significantly contribute to trial-to-trial output variability of a rate signal. However, it does not impair the temporal accuracy of signal integration. We conclude that signals from intermediate-range projections onto neocortical neurons are propagated and integrated in a highly reliable and precise manner, and may serve as a substrate for temporally precise signal transmission in neocortical networks.