Predicting the activity phase of a follower neuron with A-current in an inhibitory network

The transient potassium A-current is present in most neurons and plays an important role in determining the timing of action potentials. We examine the role of the A-current in the activity phase of a follower neuron in a rhythmic feed-forward inhibitory network with a reduced three-variable model and conduct experiments to verify the usefulness of our model. Using geometric analysis of dynamical systems, we explore the factors that determine the onset of activity in a follower neuron following release from inhibition. We first analyze the behavior of the follower neuron in a single cycle and find that the phase plane structure of the model can be used to predict the potential behaviors of the follower neuron following release from inhibition. We show that, depending on the relative scales of the inactivation time constant of the A-current and the time constant of the recovery variable, the follower neuron may or may not reach its active state following inhibition. Our simple model is used to derive a recursive set of equations to predict the contribution of the A-current parameters in determining the activity phase of a follower neuron as a function of the duration and frequency of the inhibitory input it receives. These equations can be used to demonstrate the dependence of activity phase on the period and duty cycle of the periodic inhibition, as seen by comparing the predictions of the model with the activity of the pyloric constrictor (PY) neurons in the crustacean pyloric network.     

Optimal stimulus shapes for neuronal excitation

An important problem in neuronal computation is to discern how features of stimuli control the timing of action potentials. One aspect of this problem is to determine how an action potential, or spike, can be elicited with the least energy cost, e.g., a minimal amount of applied current. Here we show in the Hodgkin & Huxley model of the action potential and in experiments on squid giant axons that: 1) spike generation in a neuron can be highly discriminatory for stimulus shape and 2) the optimal stimulus shape is dependent upon inputs to the neuron. We show how polarity and time course of post-synaptic currents determine which of these optimal stimulus shapes best excites the neuron. These results are obtained mathematically using the calculus of variations and experimentally using a stochastic search methodology. Our findings reveal a surprising complexity of computation at the single cell level that may be relevant for understanding optimization of signaling in neurons and neuronal networks.     

Multineuronal firing patterns in the signal from eye to brain

Population codes in the brain have generally been characterized by recording responses from one neuron at a time. This approach will miss codes that rely on concerted patterns of action potentials from many cells. Here we analyze visual signaling in populations of ganglion cells recorded from the isolated salamander retina. These neurons tend to fire synchronously far more frequently than expected by chance. We present an efficient algorithm to identify what groups of cells cooperate in this way. Such groups can include up to seven or more neurons and may account for more than 50% of all the spikes recorded from the retina. These firing patterns represent specific messages about the visual stimulus that differ significantly from what one would derive by single-cell analysis.     

Spike-timing dependent synaptic plasticity: a phenomenological framework

 In this paper a phenomenological model of spike-timing dependent synaptic plasticity (STDP) is developed that is based on a Volterra series-like expansion. Synaptic weight changes as a function of the relative timing of pre- and postsynaptic spikes are described by integral kernels that can easily be inferred from experimental data. The resulting weight dynamics can be stated in terms of statistical properties of pre- and postsynaptic spike trains. Generalizations to neurons that fire two different types of action potentials, such as cerebellar Purkinje cells where synaptic plasticity depends on correlations in two distinct presynaptic fibers, are discussed. We show that synaptic plasticity, together with strictly local bounds for the weights, can result in synaptic competition that is required for any form of pattern formation. This is illustrated by a concrete example where a single neuron equipped with STDP can selectively strengthen those synapses with presynaptic neurons that reliably deliver precisely timed spikes at the expense of other synapses which transmit spikes with a broad temporal distribution. Such a mechanism may be of vital importance for any neuronal system where information is coded in the timing of individual action potentials. Received: 23 January 2002 / Accepted: 28 March 2002 Correspondence to: W.M. Kistler (e-mail: kistler@anat.fgg.eur.nl Fax: +31 10 408 5459)     

Computing Complex Visual Features with Retinal Spike Times

Neurons in sensory systems can represent information not only by their firing rate, but also by the precise timing of individual spikes. For example, certain retinal ganglion cells, first identified in the salamander, encode the spatial structure of a new image by their first-spike latencies. Here we explore how this temporal code can be used by downstream neural circuits for computing complex features of the image that are not available from the signals of individual ganglion cells. To this end, we feed the experimentally observed spike trains from a population of retinal ganglion cells to an integrate-and-fire model of post-synaptic integration. The synaptic weights of this integration are tuned according to the recently introduced tempotron learning rule. We find that this model neuron can perform complex visual detection tasks in a single synaptic stage that would require multiple stages for neurons operating instead on neural spike counts. Furthermore, the model computes rapidly, using only a single spike per afferent, and can signal its decision in turn by just a single spike. Extending these analyses to large ensembles of simulated retinal signals, we show that the model can detect the orientation of a visual pattern independent of its phase, an operation thought to be one of the primitives in early visual processing. We analyze how these computations work and compare the performance of this model to other schemes for reading out spike-timing information. These results demonstrate that the retina formats spatial information into temporal spike sequences in a way that favors computation in the time domain. Moreover, complex image analysis can be achieved already by a simple integrate-and-fire model neuron, emphasizing the power and plausibility of rapid neural computing with spike times.     

Long-term activity-dependent plasticity of action potential propagation delay and amplitude in cortical networks

Background

The precise temporal control of neuronal action potentials is essential for regulating many brain functions. From the viewpoint of a neuron, the specific timings of afferent input from the action potentials of its synaptic partners determines whether or not and when that neuron will fire its own action potential. Tuning such input would provide a powerful mechanism to adjust neuron function and in turn, that of the brain. However, axonal plasticity of action potential timing is counter to conventional notions of stable propagation and to the dominant theories of activity-dependent plasticity focusing on synaptic efficacies.

Methodology/Principal Findings

Here we show the occurrence of activity-dependent plasticity of action potential propagation delays (up to 4 ms or 40% after minutes and 13 ms or 74% after hours) and amplitudes (up to 87%). We used a multi-electrode array to induce, detect, and track changes in propagation in multiple neurons while they adapted to different patterned stimuli in controlled neocortical networks in vitro. The changes did not occur when the same stimulation was repeated while blocking ionotropic gabaergic and glutamatergic receptors. Even though induction of changes in action potential timing and amplitude depended on synaptic transmission, the expression of these changes persisted in the presence of the synaptic receptor blockers.

Conclusions/Significance

We conclude that, along with changes in synaptic efficacy, propagation plasticity provides a cellular mechanism to tune neuronal network function in vitro and potentially learning and memory in the brain.     

Estimation of the individual firing frequencies of two neurons recorded with a single electrode

When monitoring neurons with a single extracellular electrode, it is common to record action potentials from different neurons. A recurring problem with such recordings is to identify which neuron is active. Sorting spikes into separate classes is possible if each neuron discharge spikes differing by their shapes and sizes. However, this approach is not applicable when the spikes are indistinguishable. In this paper, we develop a method for estimating the respective firing frequencies of two neurons, producing indistinguishable spikes. It is based on the fact that, when a neuron fires a spike, there is an interval of time during which the probability of generating a second spike is very low. If a spike occurs during this 'silent period', it is likely to be generated from another neuron and the number of occurrences of such 'doublets' can be used to estimate the respective frequencies of two spike trains. We demonstrate here that a simple relation holds between the frequency of doublets d, the respective frequencies of the two neurons A and B, fA and fB, and a chosen value Delta shorter than the silent period, d=2fAfBDelta. This relation holds for a wide class of firing processes. We used this method to analyze responses from Drosophila taste sensilla. We first checked if the method was consistent with results obtained with stimuli that elicit responses of two taste neurons firing distinguishable spikes. We then applied this method to the study of a pair of taste neurons involved in the coding for salt taste in Drosophila melanogaster.     

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein–Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein–Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.     

Electrical Interactions via the Extracellular Potential Near Cell Bodies

Ephaptic interactions between a neuron and axons or dendrites passing by its cell body can be, in principle, more significant than ephaptic interactions among axons in a fiber tract. Extracellular action potentials outside axons are small in amplitude and spatially spread out, while they are larger in amplitude and much more spatially confined near cell bodies. We estimated the extracellular potentials associated with an action potential in a cortical pyramidal cell using standard one-dimensional cable theory and volume conductor theory. Their spatial and temporal pattern reveal much about the location and timing of currents in the cell, especially in combination with a known morphology, and simple experiments could resolve questions about spike initiation. From the extracellular potential we compute the ephaptically induced polarization in a nearby passive cable. The magnitude of this induced voltage can be several mV, does not spread electrotonically, and depends only weakly on the passive properties of the cable. We discuss their possible functional relevance.     

Role of an A-Type K+ Conductance in the Back-Propagation of Action Potentials in the Dendrites of Hippocampal Pyramidal Neurons

Action potentials elicited in the axon actively back-propagate into the dendritic tree. During this process their amplitudes can be modulated by internal and external factors. We used a compartmental model of a hippocampal CA1 pyramidal neuron to illustrate how this modulation could depend on (1) the properties of an A-type K⁺ conductance that is expressed at high density in hippocampal dendrites and (2) the relative timing of synaptic activation. The simulations suggest that the time relationship between pre- and postsynaptic activity could help regulate the amplitude of back-propagating action potentials, especially in the distal portion of the dendritic tree.     

Pre & postsynaptic tuning of action potential timing by spontaneous GABAergic activity

Frequency and timing of action potential discharge are key elements for coding and transfer of information between neurons. The nature and location of the synaptic contacts, the biophysical parameters of the receptor-operated channels and their kinetics of activation are major determinants of the firing behaviour of each individual neuron. Ultimately the intrinsic excitability of each neuron determines the input-output function. Here we evaluate the influence of spontaneous GABAergic synaptic activity on the timing of action potentials in Layer 2/3 pyramidal neurones in acute brain slices from the somatosensory cortex of young rats. Somatic dynamic current injection to mimic synaptic input events was employed, together with a simple computational model that reproduce subthreshold membrane properties. Besides the well-documented control of neuronal excitability, spontaneous background GABAergic activity has a major detrimental effect on spike timing. In fact, GABA(A) receptors tune the relationship between the excitability and fidelity of pyramidal neurons via a postsynaptic (the reversal potential for GABA(A) activity) and a presynaptic (the frequency of spontaneous activity) mechanism. GABAergic activity can decrease or increase the excitability of pyramidal neurones, depending on the difference between the reversal potential for GABA(A) receptors and the threshold for action potential. In contrast, spike time jitter can only be increased proportionally to the difference between these two membrane potentials. Changes in excitability by background GABAergic activity can therefore only be associated with deterioration of the reliability of spike timing.     

To Burst or Not to Burst?

It is well known that some neurons tend to fire packets of action potentials followed by periods of quiescence (bursts) while others within the same stage of sensory processing fire in a tonic manner. However, the respective computational advantages of bursting and tonic neurons for encoding time varying signals largely remain a mystery. Weakly electric fish use cutaneous electroreceptors to convey information about sensory stimuli and it has been shown that some electroreceptors exhibit bursting dynamics while others do not. In this study, we compare the neural coding capabilities of tonically firing and bursting electroreceptor model neurons using information theoretic measures. We find that both bursting and tonically firing model neurons efficiently transmit information about the stimulus. However, the decoding mechanisms that must be used for each differ greatly: a non-linear decoder would be required to extract all the available information transmitted by the bursting model neuron whereas a linear one might suffice for the tonically firing model neuron. Further investigations using stimulus reconstruction techniques reveal that, unlike the tonically firing model neuron, the bursting model neuron does not encode the detailed time course of the stimulus. A novel measure of feature detection reveals that the bursting neuron signals certain stimulus features. Finally, we show that feature extraction and stimulus estimation are mutually exclusive computations occurring in bursting and tonically firing model neurons, respectively. Our results therefore suggest that stimulus estimation and feature extraction might be parallel computations in certain sensory systems rather than being sequential as has been previously proposed.     

Ion channel noise can explain firing correlation in auditory nerves

Sensory neurons code information about stimuli in their sequence of action potentials (spikes). Intuitively, the spikes should represent stimuli with high fidelity. However, generating and propagating spikes is a metabolically expensive process. It is therefore likely that neural codes have been selected to balance energy expenditure against encoding error. Our recently proposed optimal, energy-constrained neural coder (Jones et al. Frontiers in Computational Neuroscience, 9, 61 2015) postulates that neurons time spikes to minimize the trade-off between stimulus reconstruction error and expended energy by adjusting the spike threshold using a simple dynamic threshold. Here, we show that this proposed coding scheme is related to existing coding schemes, such as rate and temporal codes. We derive an instantaneous rate coder and show that the spike-rate depends on the signal and its derivative. In the limit of high spike rates the spike train maximizes fidelity given an energy constraint (average spike-rate), and the predicted interspike intervals are identical to those generated by our existing optimal coding neuron. The instantaneous rate coder is shown to closely match the spike-rates recorded from P-type primary afferents in weakly electric fish. In particular, the coder is a predictor of the peristimulus time histogram (PSTH). When tested against in vitro cortical pyramidal neuron recordings, the instantaneous spike-rate approximates DC step inputs, matching both the average spike-rate and the time-to-first-spike (a simple temporal code). Overall, the instantaneous rate coder relates optimal, energy-constrained encoding to the concepts of rate-coding and temporal-coding, suggesting a possible unifying principle of neural encoding of sensory signals.     

Spike timing precision of neuronal circuits

Spike timing is believed to be a key factor in sensory information encoding and computations performed by the neurons and neuronal circuits. However, the considerable noise and variability, arising from the inherently stochastic mechanisms that exist in the neurons and the synapses, degrade spike timing precision. Computational modeling can help decipher the mechanisms utilized by the neuronal circuits in order to regulate timing precision. In this paper, we utilize semi-analytical techniques, which were adapted from previously developed methods for electronic circuits, for the stochastic characterization of neuronal circuits. These techniques, which are orders of magnitude faster than traditional Monte Carlo type simulations, can be used to directly compute the spike timing jitter variance, power spectral densities, correlation functions, and other stochastic characterizations of neuronal circuit operation. We consider three distinct neuronal circuit motifs: Feedback inhibition, synaptic integration, and synaptic coupling. First, we show that both the spike timing precision and the energy efficiency of a spiking neuron are improved with feedback inhibition. We unveil the underlying mechanism through which this is achieved. Then, we demonstrate that a neuron can improve on the timing precision of its synaptic inputs, coming from multiple sources, via synaptic integration: The phase of the output spikes of the integrator neuron has the same variance as that of the sample average of the phases of its inputs. Finally, we reveal that weak synaptic coupling among neurons, in a fully connected network, enables them to behave like a single neuron with a larger membrane area, resulting in an improvement in the timing precision through cooperation.     

The morphological basis of intracellular measurements in the cockroach tactile spine neuron

Summary The femoral tactile spine of the cockroach (Periplaneta americana) contains a single sensory neuron, which adapts rapidly and completely to step deformations of the spine. Techniques for stable intracellular recording from the tactile spine neuron have recently been established, allowing electrophysiological investigation of mechanotransduction and adaptation in this sensory neuron. However, intracellular recordings from the neuron produce a wide range of action potential heights and thresholds, raising the possibility that some penetrations are in adjacent, but closely coupled supporting glial cells. This problem is exacerbated because the cell cannot be visualized during penetration.Systematic measurements of action potential heights and thresholds were made in tactile spine cells, together with identification of some penetrated cells by intracellular injection of Lucifer Yellow. All stained cells were clearly sensory neurons, although their action potentials amplitudes varied from 9 mV to 80 mV. Smaller action potentials were broader than larger action potentials, and the changes in height and shape could be explained by a simple cable conduction model using measured morphological and electrical parameters. The model could also account for the observed relationship between action potential height and threshold.These results indicate that reliable recording from the tactile spine neuron is possible, but that variability in the positions of the penetration or the spike initiating zone cause an apparently wide range of electrophysiological measurements.     

Temporal encoding in nervous systems: A rigorous definition

We propose a rigorous definition for the termtemporal encoding as it is applied to schemes for the representation of information withinpatterns of neuronal action potentials, and distinguish temporal encoding schemes from those based on window-averagedmean rate encoding. The definition relies on the identification of anencoding time window, defined as the duration of a neuron's spike train assumed to correspond to a single symbol in the neural code. The duration of the encoding time window is dictated by the time scale of the information being encoded. We distinguish between the concepts of theencoding time window and theintegration time window, the latter of which is defined as the duration of a stimulus signal that affects the response of the neuron. We note that the duration of the encoding and integration windows might be significantly different. We also present objective, experimentally assessable criteria for identifying neurons and neuronal ensembles that utilize temporal encoding to any significant extent. The definitions and criteria are made rigorous within the contexts of several commonly used analytical approaches, including thestimulus reconstruction analysis technique. Several examples are presented to illustrate the distinctions between and relative capabilities of rate encoding and temporal encoding schemes. We also distinguish our usage oftemporal encoding from the termtemporal coding, which is commonly used in reference to the representation of information about thetiming of events by rate encoding schemes.     

Ion channel density and threshold dynamics of repetitive firing in a cortical neuron model

Modifying the density and distribution of ion channels in a neuron (by natural up- and down-regulation, by pharmacological intervention or by spontaneous mutations) changes its activity pattern. In the present investigation, we analyze how the impulse patterns are regulated by the density of voltage-gated channels in a model neuron, based on voltage clamp measurements of hippocampal interneurons. At least three distinct oscillatory patterns, associated with three distinct regions in the Na-K channel density plane, were found. A stability analysis showed that the different regions are characterized by saddle-node, double-orbit, and Hopf bifurcation threshold dynamics, respectively. Single strongly graded action potentials occur in an area outside the oscillatory regions, but less graded action potentials occur together with repetitive firing over a considerable range of channel densities. The presently found relationship between channel densities and oscillatory behavior may be relevance for understanding principal spiking patterns of cortical neurons (regular firing and fast spiking). It may also be of relevance for understanding the action of pharmacological compounds on brain oscillatory activity.     

Coding of stimulus intensity in an olfactory receptor neuron: Role of neuron spatial extent and passive dendritic backpropagation of action potentials

The olfactory receptor neuron provides a good opportunity to analyze a biophysical model of a single neuron because its dendritic structure is simple and even close to a cylinder in the case of the moth sex-pheromone receptor cell. We have considered this cylindrical case and studied two main problems. First, we were concerned with the effect of the neuron's length on the receptor potential for a constant stimulus-induced conductance change. An analytical solution for the receptor potential was determined by using input, resistances. It was shown that the longer the neuron, the greater its ability to code over a wide range of values of the intensity of the stimulus. Second, we studied numerically the passive backpropagation of action potentials into the dendrite and its influence on the firing frequency. While propagating along the dendrite the action potential decreases in amplitude and its shape becomes rounded. The firing frequency in the model with backpropagation was found to be greater than that obtained analytically in the absence of backpropagation. However, for any given conductance change, when normalized with respect to their maxima, both firing frequencies were found to be very similar over a wide range of parameter values. Therefore, the actual firing rate (with backpropagation) may be approximated by the analytical solution without backpropagation if the actual firing rate for a large conductance change is known.