Firing-rate models for neurons with a broad repertoire of spiking behaviors

Capturing the response behavior of spiking neuron models with rate-based models facilitates the investigation of neuronal networks using powerful methods for rate-based network dynamics. To this end, we investigate the responses of two widely used neuron model types, the Izhikevich and augmented multi-adapative threshold (AMAT) models, to a range of spiking inputs ranging from step responses to natural spike data. We find (i) that linear-nonlinear firing rate models fitted to test data can be used to describe the firing-rate responses of AMAT and Izhikevich spiking neuron models in many cases; (ii) that firing-rate responses are generally too complex to be captured by first-order low-pass filters but require bandpass filters instead; (iii) that linear-nonlinear models capture the response of AMAT models better than of Izhikevich models; (iv) that the wide range of response types evoked by current-injection experiments collapses to few response types when neurons are driven by stationary or sinusoidally modulated Poisson input; and (v) that AMAT and Izhikevich models show different responses to spike input despite identical responses to current injections. Together, these findings suggest that rate-based models of network dynamics may capture a wider range of neuronal response properties by incorporating second-order bandpass filters fitted to responses of spiking model neurons. These models may contribute to bringing rate-based network modeling closer to the reality of biological neuronal networks.     

A generalized linear model for estimating spectrotemporal receptive fields from responses to natural sounds

In the auditory system, the stimulus-response properties of single neurons are often described in terms of the spectrotemporal receptive field (STRF), a linear kernel relating the spectrogram of the sound stimulus to the instantaneous firing rate of the neuron. Several algorithms have been used to estimate STRFs from responses to natural stimuli; these algorithms differ in their functional models, cost functions, and regularization methods. Here, we characterize the stimulus-response function of auditory neurons using a generalized linear model (GLM). In this model, each cell's input is described by: 1) a stimulus filter (STRF); and 2) a post-spike filter, which captures dependencies on the neuron's spiking history. The output of the model is given by a series of spike trains rather than instantaneous firing rate, allowing the prediction of spike train responses to novel stimuli. We fit the model by maximum penalized likelihood to the spiking activity of zebra finch auditory midbrain neurons in response to conspecific vocalizations (songs) and modulation limited (ml) noise. We compare this model to normalized reverse correlation (NRC), the traditional method for STRF estimation, in terms of predictive power and the basic tuning properties of the estimated STRFs. We find that a GLM with a sparse prior predicts novel responses to both stimulus classes significantly better than NRC. Importantly, we find that STRFs from the two models derived from the same responses can differ substantially and that GLM STRFs are more consistent between stimulus classes than NRC STRFs. These results suggest that a GLM with a sparse prior provides a more accurate characterization of spectrotemporal tuning than does the NRC method when responses to complex sounds are studied in these neurons.     

Context-dependent coding in single neurons

The linear-nonlinear cascade model (LN model) has proven very useful in representing a neural system’s encoding properties, but has proven less successful in reproducing the firing patterns of individual neurons whose behavior is strongly dependent on prior firing history. While the cell’s behavior can still usefully be considered as feature detection acting on a fluctuating input, some of the coding capacity of the cell is taken up by the increased firing rate due to a constant “driving” direct current (DC) stimulus. Furthermore, both the DC input and the post-spike refractory period generate regular firing, reducing the spike-timing entropy available for encoding time-varying fluctuations. In this paper, we address these issues, focusing on the example of motoneurons in which an afterhyperpolarization (AHP) current plays a dominant role regularizing firing behavior. We explore the accuracy and generalizability of several alternative models for single neurons under changes in DC and variance of the stimulus input. We use a motoneuron simulation to compare coding models in neurons with and without the AHP current. Finally, we quantify the tradeoff between instantaneously encoding information about fluctuations and about the DC.     

Dynamics of the Instantaneous Firing Rate in Response to Changes in Input Statistics

We review and extend recent results on the instantaneous firing rate dynamics of simplified models of spiking neurons in response to noisy current inputs. It has been shown recently that the response of the instantaneous firing rate to small amplitude oscillations in the mean inputs depends in the large frequency limit f on the spike initiation dynamics. A particular simplified model, the exponential integrate-and-fire (EIF) model, has a response that decays as 1/f in the large frequency limit and describes very well the response of conductance-based models with a Hodgkin-Huxley type fast sodium current. Here, we show that the response of the EIF instantaneous firing rate also decays as 1/f in the case of an oscillation in the variance of the inputs for both white and colored noise. We then compute the initial transient response of the firing rate of the EIF model to a step change in its mean inputs and/or in the variance of its inputs. We show that in both cases the response speed is proportional to the neuron stationary firing rate and inversely proportional to a spike slope factor _T that controls the sharpness of spike initiation: as 1/_T for a step change in mean inputs, and as 1/_T² for a step change in the variance in the inputs.     

The Elementary Operations of Human Vision Are Not Reducible to Template-Matching

It is generally acknowledged that biological vision presents nonlinear characteristics, yet linear filtering accounts of visual processing are ubiquitous. The template-matching operation implemented by the linear-nonlinear cascade (linear filter followed by static nonlinearity) is the most widely adopted computational tool in systems neuroscience. This simple model achieves remarkable explanatory power while retaining analytical tractability, potentially extending its reach to a wide range of systems and levels in sensory processing. The extent of its applicability to human behaviour, however, remains unclear. Because sensory stimuli possess multiple attributes (e.g. position, orientation, size), the issue of applicability may be asked by considering each attribute one at a time in relation to a family of linear-nonlinear models, or by considering all attributes collectively in relation to a specified implementation of the linear-nonlinear cascade. We demonstrate that human visual processing can operate under conditions that are indistinguishable from linear-nonlinear transduction with respect to substantially different stimulus attributes of a uniquely specified target signal with associated behavioural task. However, no specific implementation of a linear-nonlinear cascade is able to account for the entire collection of results across attributes; a satisfactory account at this level requires the introduction of a small gain-control circuit, resulting in a model that no longer belongs to the linear-nonlinear family. Our results inform and constrain efforts at obtaining and interpreting comprehensive characterizations of the human sensory process by demonstrating its inescapably nonlinear nature, even under conditions that have been painstakingly fine-tuned to facilitate template-matching behaviour and to produce results that, at some level of inspection, do conform to linear filtering predictions. They also suggest that compliance with linear transduction may be the targeted outcome of carefully crafted nonlinear circuits, rather than default behaviour exhibited by basic components.     

The relationship between neuronal calcium concentration and firing rate during stochastic synaptic inputs

We propose a novel, nonlinear theory about reading neuronal information using intracellular calcium concentrations, which includes the linear theory already developed in the literature as a special case. The theory is numerically confirmed using the Pinsky-Rinzel and integrate-and-fire models with constant rate Poisson inputs. Applying the theory to models with non-constant inputs, we find that there is a time lag equal to the calcium buffering time constant between the instantaneous firing rate and the firing rate estimated using calcium concentrations.     

Differences in olfactory bulb mitral cell spiking with ortho- and retronasal stimulation revealed by data-driven models

The majority of olfaction studies focus on orthonasal stimulation where odors enter via the front nasal cavity, while retronasal olfaction, where odors enter the rear of the nasal cavity during feeding, is understudied. The coding of retronasal odors via coordinated spiking of neurons in the olfactory bulb (OB) is largely unknown despite evidence that higher level processing is different than orthonasal. To this end, we use multi-electrode array in vivo recordings of rat OB mitral cells (MC) in response to a food odor with both modes of stimulation, and find significant differences in evoked firing rates and spike count covariances (i.e., noise correlations). Differences in spiking activity often have implications for sensory coding, thus we develop a single-compartment biophysical OB model that is able to reproduce key properties of important OB cell types. Prior experiments in olfactory receptor neurons (ORN) showed retro stimulation yields slower and spatially smaller ORN inputs than with ortho, yet whether this is consequential for OB activity remains unknown. Indeed with these specifications for ORN inputs, our OB model captures the salient trends in our OB data. We also analyze how first and second order ORN input statistics dynamically transfer to MC spiking statistics with a phenomenological linear-nonlinear filter model, and find that retro inputs result in larger linear filters than ortho inputs. Finally, our models show that the temporal profile of ORN is crucial for capturing our data and is thus a distinguishing feature between ortho and retro stimulation, even at the OB. Using data-driven modeling, we detail how ORN inputs result in differences in OB dynamics and MC spiking statistics. These differences may ultimately shape how ortho and retro odors are coded.     

Firing-rate models capture essential response dynamics of LGN relay cells

Firing-rate models provide a practical tool for studying signal processing in the early visual system, permitting more thorough mathematical analysis than spike-based models. We show here that essential response properties of relay cells in the lateral geniculate nucleus (LGN) can be captured by surprisingly simple firing-rate models consisting of a low-pass filter and a nonlinear activation function. The starting point for our analysis are two spiking neuron models based on experimental data: a spike-response model fitted to data from macaque (Carandini et al. J. Vis., 20(14), 1–2011, 2007), and a model with conductance-based synapses and afterhyperpolarizing currents fitted to data from cat (Casti et al. J. Comput. Neurosci., 24(2), 235–252, 2008). We obtained the nonlinear activation function by stimulating the model neurons with stationary stochastic spike trains, while we characterized the linear filter by fitting a low-pass filter to responses to sinusoidally modulated stochastic spike trains. To account for the non-Poisson nature of retinal spike trains, we performed all analyses with spike trains with higher-order gamma statistics in addition to Poissonian spike trains. Interestingly, the properties of the low-pass filter depend only on the average input rate, but not on the modulation depth of sinusoidally modulated input. Thus, the response properties of our model are fully specified by just three parameters (low-frequency gain, cutoff frequency, and delay) for a given mean input rate and input regularity. This simple firing-rate model reproduces the response of spiking neurons to a step in input rate very well for Poissonian as well as for non-Poissonian input. We also found that the cutoff frequencies, and thus the filter time constants, of the rate-based model are unrelated to the membrane time constants of the underlying spiking models, in agreement with similar observations for simpler models.     

Firing patterns in the adaptive exponential integrate-and-fire model

For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.     

Modeling the Afferent Dynamics of the Baroreflex Control System

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods.     

Variable synaptic strengths controls the firing rate distribution in feedforward neural networks

Heterogeneity of firing rate statistics is known to have severe consequences on neural coding. Recent experimental recordings in weakly electric fish indicate that the distribution-width of superficial pyramidal cell firing rates (trial- and time-averaged) in the electrosensory lateral line lobe (ELL) depends on the stimulus, and also that network inputs can mediate changes in the firing rate distribution across the population. We previously developed theoretical methods to understand how two attributes (synaptic and intrinsic heterogeneity) interact and alter the firing rate distribution in a population of integrate-and-fire neurons with random recurrent coupling. Inspired by our experimental data, we extend these theoretical results to a delayed feedforward spiking network that qualitatively capture the changes of firing rate heterogeneity observed in in-vivo recordings. We demonstrate how heterogeneous neural attributes alter firing rate heterogeneity, accounting for the effect with various sensory stimuli. The model predicts how the strength of the effective network connectivity is related to intrinsic heterogeneity in such delayed feedforward networks: the strength of the feedforward input is positively correlated with excitability (threshold value for spiking) when firing rate heterogeneity is low and is negatively correlated with excitability with high firing rate heterogeneity. We also show how our theory can be used to predict effective neural architecture. We demonstrate that neural attributes do not interact in a simple manner but rather in a complex stimulus-dependent fashion to control neural heterogeneity and discuss how it can ultimately shape population codes.     

Universality in neural networks: the importance of the ‘mean firing rate’

We present a general analysis of highly connected recurrent neural networks which are able to learn and retrieve a finite number of static patterns. The arguments are based on spike trains and their interval distribution and require no specific model of a neuron. In particular, they apply to formal two-state neurons as well as to more refined models like the integrate-and-fire neuron or the Hodgkin-Huxley equations. We show that the mean firing rate defined as the inverse of the mean interval length is the only relevant parameter (apart from the synaptic weights) that determines the existence of retrieval solutions with a large overlap with one of the learnt patterns. The statistics of the spiking noise (Gaussian, Poisson or other) and hence the shape of the interval distribution does not matter. Thus our unifying approach explains why, and when, all the different associative networks which treat static patterns yield basically the same results, i.e., belong to the same universality class.     

Computational simulation of the input-output relationship in hippocampal pyramidal cells

The precise mapping of how complex patterns of synaptic inputs are integrated into specific patterns of spiking output is an essential step in the characterization of the cellular basis of network dynamics and function. Relative to other principal neurons of the hippocampus, the electrophysiology of CA1 pyramidal cells has been extensively investigated. Yet, the precise input-output relationship is to date unknown even for this neuronal class. CA1 pyramidal neurons receive laminated excitatory inputs from three distinct pathways: recurrent CA1 collaterals on basal dendrites, CA3 Schaffer collaterals, mostly on oblique and proximal apical dendrites, and entorhinal perforant pathway on distal apical dendrites. We implemented detailed computer simulations of pyramidal cell electrophysiology based on three-dimensional anatomical reconstructions and compartmental models of available biophysical properties from the experimental literature. To investigate the effect of synaptic input on axosomatic firing, we stochastically distributed a realistic number of excitatory synapses in each of the three dendritic layers. We then recorded the spiking response to different stimulation patterns. For all dendritic layers, synchronous stimuli resulted in trains of spiking output and a linear relationship between input and output firing frequencies. In contrast, asynchronous stimuli evoked non-bursting spike patterns and the corresponding firing frequency input-output function was logarithmic. The regular/irregular nature of the input synaptic intervals was only reflected in the regularity of output inter-burst intervals in response to synchronous stimulation, and never affected firing frequency. Synaptic stimulations in the basal and proximal apical trees across individual neuronal morphologies yielded remarkably similar input-output relationships. Results were also robust with respect to the detailed distributions of dendritic and synaptic conductances within a plausible range constrained by experimental evidence. In contrast, the input-output relationship in response to distal apical stimuli showed dramatic differences from the other dendritic locations as well as among neurons, and was more sensible to the exact channel densities. Action Editor: Alain Destexhe     

Linear and quadratic models of point process systems: Contributions of patterned input to output

In the 1880's Volterra characterised a nonlinear system using a functional series connecting continuous input and continuous output. Norbert Wiener, in the 1940's, circumvented problems associated with the application of Volterra series to physical problems by deriving from it a new series of terms that are mutually uncorrelated with respect to Gaussian processes. Subsequently, Brillinger, in the 1970's, introduced a point-process analogue of Volterra's series connecting point-process inputs to the instantaneous rate of point-process output. We derive here a new series from this analogue in which its terms are mutually uncorrelated with respect to Poisson processes. This new series expresses how patterned input in a spike train, represented by third-order cross-cumulants, is converted into the instantaneous rate of an output point-process. Given experimental records of suitable duration, the contribution of arbitrary patterned input to an output process can, in principle, be determined. Solutions for linear and quadratic point-process models with one and two inputs and a single output are investigated. Our theoretical results are applied to isolated muscle spindle data in which the spike trains from the primary and secondary endings from the same muscle spindle are recorded in response to stimulation of one and then two static fusimotor axons in the absence and presence of a random length change imposed on the parent muscle. For a fixed mean rate of input spikes, the analysis of the experimental data makes explicit which patterns of two input spikes contribute to an output spike.     

Instantaneous Non-Linear Processing by Pulse-Coupled Threshold Units

Contemporary theory of spiking neuronal networks is based on the linear response of the integrate-and-fire neuron model derived in the diffusion limit. We find that for non-zero synaptic weights, the response to transient inputs differs qualitatively from this approximation. The response is instantaneous rather than exhibiting low-pass characteristics, non-linearly dependent on the input amplitude, asymmetric for excitation and inhibition, and is promoted by a characteristic level of synaptic background noise. We show that at threshold the probability density of the potential drops to zero within the range of one synaptic weight and explain how this shapes the response. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled networks of threshold units.     

Dynamics of the exponential integrate-and-fire model with slow currents and adaptation

In order to properly capture spike-frequency adaptation with a simplified point-neuron model, we study approximations of Hodgkin-Huxley (HH) models including slow currents by exponential integrate-and-fire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPA-type conductance pulses using the current-voltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIF-type models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spike-frequency adaptation, phase-response curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in time-periodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIF-type networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPA-like synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HH-type model. Our work shows that the EIF model with additional slow currents is well suited for use in large-scale, point-neuron models in which spike-frequency adaptation is important.     

The chronotron: a neuron that learns to fire temporally precise spike patterns

In many cases, neurons process information carried by the precise timings of spikes. Here we show how neurons can learn to generate specific temporally precise output spikes in response to input patterns of spikes having precise timings, thus processing and memorizing information that is entirely temporally coded, both as input and as output. We introduce two new supervised learning rules for spiking neurons with temporal coding of information (chronotrons), one that provides high memory capacity (E-learning), and one that has a higher biological plausibility (I-learning). With I-learning, the neuron learns to fire the target spike trains through synaptic changes that are proportional to the synaptic currents at the timings of real and target output spikes. We study these learning rules in computer simulations where we train integrate-and-fire neurons. Both learning rules allow neurons to fire at the desired timings, with sub-millisecond precision. We show how chronotrons can learn to classify their inputs, by firing identical, temporally precise spike trains for different inputs belonging to the same class. When the input is noisy, the classification also leads to noise reduction. We compute lower bounds for the memory capacity of chronotrons and explore the influence of various parameters on chronotrons' performance. The chronotrons can model neurons that encode information in the time of the first spike relative to the onset of salient stimuli or neurons in oscillatory networks that encode information in the phases of spikes relative to the background oscillation. Our results show that firing one spike per cycle optimizes memory capacity in neurons encoding information in the phase of firing relative to a background rhythm.     

Neuron firing in driven nonlinear integrate-and-fire models

Statistical properties of neuron firing are studied in the framework of a nonlinear leaky integrate-and-fire model that is driven by a slow periodic subthreshold signal. The firing events are characterized by first passage time densities. The experimentally better accessible interspike interval density generally depends on the sojourn times in a refractory state of the neuron. This aspect is not part of the integrate-and-fire model and must be modelled additionally. For a large class of refractory dynamics, a general expression for the interspike interval density is given and further evaluated for the two cases with an instantaneous resetting (i.e. no refractory state) and a refractory state possessing a deterministic lifetime. First passage time densities and interspike interval densities following from the proposed theory compare favorably with precise numerical simulations.