Electrical stimulation of neural tissue modeled as a cellular composite: point source electrode in an isotropic tissue

Standard volume conductor models of neural electrical stimulation assume that the electrical properties of the tissue are well described by a conductivity that is smooth and homogeneous at a microscopic scale. However, neural tissue is composed of tightly packed cells whose membranes have markedly different electrical properties to either the intra- or extracellular space. Consequently, the electrical properties of tissue are highly heterogeneous at the microscopic scale: a fact not accounted for in standard volume conductor models. Here we apply a recently developed framework for volume conductor models that accounts for the cellular composition of tissue. We consider the case of a point source electrode in tissue comprised of neural fibers crossing each other equally in all directions. We derive the tissue admittivity (that replaces the standard tissue conductivity) from single cell properties, and then calculate the extracellular potential. Our findings indicate that the cellular composition of tissue affects the spatiotemporal profile of the extracellular potential. In particular, the full solution asymptotically approaches a near-field limit close to the electrode and a far-field limit far from the electrode. The near-field and far-field approximations are solutions to standard volume conductor models, but differ from each other by nearly an order or magnitude. Consequently the full solution is expected to provide a more accurate estimate of electrical potentials over the full range of electrode-neurite separations.

Modelling extracellular electrical stimulation: part 4. Effect of the cellular composition of neural tissue on its spatio-temporal filtering properties

OBJECTIVE

The objective of this paper is to present a concrete application of the cellular composite model for calculating the membrane potential, described in an accompanying paper.

APPROACH

A composite model that is used to determine the membrane potential for both longitudinal and transverse modes of stimulation is demonstrated.

MAIN RESULTS

Two extreme limits of the model, near-field and far-field for an electrode close to or distant from a neuron, respectively, are derived in this paper. Results for typical neural tissue are compared using the composite, near-field and far-field models as well as the standard isotropic volume conductor model. The self-consistency of the composite model, its spatial profile response and the extracellular potential time behaviour are presented. The magnitudes of the longitudinal and transverse components for different values of electrode-neurite separations are compared.

SIGNIFICANCE

The unique features of the composite model and its simplified versions can be used to accurately estimate the spatio-temporal response of neural tissue to extracellular electrical stimulation.

An all-diamond, hermetic electrical feedthrough array for a retinal prosthesis

The interface between medical implants and the human nervous system is rapidly becoming more and more complex. This rise in complexity is driving the need for increasing numbers of densely packed electrical feedthrough to carry signals to and from implanted devices. This is particularly crucial in the field of neural prosthesis where high resolution stimulating or recording arrays near peripheral nerves or in the brain could dramatically improve the performance of these devices. Here we describe a flexible strategy for implementing high density, high count arrays of hermetic electrical feedthroughs by forming conducting nitrogen doped nanocrystalline diamond channels within an insulating polycrystalline diamond substrate. A unique feature of these arrays is that the feedthroughs can themselves be used as stimulating electrodes for neural tissue. Our particular application is such a feedthrough, designed as a component of a retinal implant to restore vision to the blind. The hermeticity of the feedthroughs means that the array can also form part of an implantable capsule which can interface directly with internal electronic chips. The hermeticity of the array is demonstrated by helium leak tests and electrical and electrochemical characterisation of the feedthroughs is described. The nitrogen doped nanocrystalline diamond forming the electrical feedthroughs is shown to be non-cyctotoxic. New fabrication strategies, such as the one described here, combined with the exceptional biostability of diamond can be exploited to generate a range of biomedical implants that last for the lifetime of the user without fear of degradation.

Multicompartment retinal ganglion cells response to high frequency bi-phasic pulse train stimulation: Simulation results

Retinal ganglion cells (RGCs) are the sole output neurons of the retina that carry information about a visual scene to the brain. By stimulating RGCs with electrical stimulation, it is possible to elicit a sensation of light for people with macular degeneration or retinitis pigmentosa. To investigate the responses of RGCs to high frequency bi-phasic pulse train stimulation, we use previously constrained models of multi-compartment OFF RGCs. The morphologies of mouse RGCs are taken from the Chalupa set of the NeuroMorpho database. The cell models are divided into compartments representing the dendrites, soma and axon that vary between the cells. A total of 132 cells are simulated in the NEURON environment. Results show that the cell morphology plays an important role in the response characteristics of the cell to high frequency bi-phasic pulse train stimulation.

Retinal ganglion cells electrophysiology: the effect of cell morphology on impulse waveform

There are 16 morphologically defined classes of rats retinal ganglion cells (RGCs). Using computer simulation of a realistic anatomically correct A1 mouse RGC, we investigate the effect of the cell's morphology on its impulse waveform, using the first-, and second-order time derivatives as well as the phase plot features. Using whole cell patch clamp recordings, we recorded the impulse waveform for each of the rat RGCs types. While we found some clear differences in many features of the impulse waveforms for A2 and B2 cells compared to other cell classes, many cell types did not show clear differences.

Internal inconsistencies in models of electrical stimulation in neural tissue

Calculating the membrane potential of a neurite under extracellular electrical stimulation is important in the design of some recent stimulation strategies for neuroprosthetic devices including retinal implants, cochlear implants, deep brain stimulation. A common approach, widely used in the electrical stimulation literature uses a volume conductor model to calculate the electrical potential in the tissue and then extracts the voltage or current density on the surface of a neuron, which is used as input to the cable equation to calculate the neuron's response. However this approach ignores the effect of the neuron itself as well as surrounding neurons on the extracellular potential. Here we highlight that this leads to an internal inconsistency in the overall model because the result depends on whether the voltage or current density is used to calculate the neural response. The magnitude of this discrepancy is calculated for the example of a point source electrode in a homogeneous medium and is shown to be up to several hundred percent under some stimulus conditions. The inconsistency can be resolved by ensuring that the voltage is related to the current density by the transimpedance of the neurite. Deriving a volume conductor model that satisfies this relationship requires further work.

Determining the electrical impedance of the retina from a complex voltage map

A method for determining the electrical impedance of the retina from complex valued voltage measurements is described. This is an extension to our previous work which did not consider the permittivity of the retina, or an exploration of the number of voltage measurements required to form an adequate solution. The model considers inhomogeneity in permittivity and conductivity of the tissue in both the x and y directions. This framework is tested on noisy voltage data solved for a synthetic rectangular retinal section. A synthetic retinal section inclusive of fovea is also considered. Estimates of the conductivity and permittivity maps are solved for using finite element analysis for a range of down sampling factors and signal to noise ratios for the voltage data. This is done to assess the potential accuracy of this method in a physical experiment.

Minimisation of required charge for desired neuronal spike rate

Retinal implants restore limited visual perception to blind implantees by electrical stimulation of surviving neurons. We consider the efficacy of two electrical stimulation parameters, frequency of stimulation and interphase gap between cathodic and anodic phases, on the required charge to reach a desired neuronal spike rate. Using a Hodgkin-Huxley model of a neuron, we find the most efficient means of achieving a desired spike rate for neurons by electrical stimulation is to use a stimulation frequency identical to the desired spike rate, as well as a long interphase gap.

Epiretinal electrical stimulation and the inner limiting membrane in rat retina

In this paper we aim to quantify the effect of the inner limiting membrane (ILM) of the retina on the thresholds for epiretinal electrical stimulation of retinal ganglion cells by a microelectronic retinal prosthesis. A pair of bipolar stimulating electrodes was placed either above (on the epiretinal surface) or below the ILM while we made whole-cell patch-clamp recordings from retinal ganglion cells in an isolated rat retinal whole-mount preparation. Across our cell population we found no significant difference in the median threshold stimulus amplitudes when the stimulating electrodes were placed below as opposed to above the ILM (p = 0.08). However, threshold stimulus amplitudes did tend to be lower when the stimulating electrodes were placed below the ILM (30 µA vs 56 µA).

The effect of morphology upon electrophysiological responses of retinal ganglion cells: simulation results

Retinal ganglion cells (RGCs) display differences in their morphology and intrinsic electrophysiology. The goal of this study is to characterize the ionic currents that explain the behavior of ON and OFF RGCs and to explore if all morphological types of RGCs exhibit the phenomena described in electrophysiological data. We extend our previous single compartment cell models of ON and OFF RGCs to more biophysically realistic multicompartment cell models and investigate the effect of cell morphology on intrinsic electrophysiological properties. The membrane dynamics are described using the Hodgkin - Huxley type formalism. A subset of published patch-clamp data from isolated intact mouse retina is used to constrain the model and another subset is used to validate the model. Two hundred morphologically distinct ON and OFF RGCs are simulated with various densities of ionic currents in different morphological neuron compartments. Our model predicts that the differences between ON and OFF cells are explained by the presence of the low voltage activated calcium current in OFF cells and absence of such in ON cells. Our study shows through simulation that particular morphological types of RGCs are capable of exhibiting the full range of phenomena described in recent experiments. Comparisons of outputs from different cells indicate that the RGC morphologies that best describe recent experimental results are ones that have a larger ratio of soma to total surface area.

Information theoretic inference of the optimal number of electrodes for future cochlear implants using a spiral cochlea model

Contemporary cochlear implants stimulate the auditory nerve with an array of up to 22 electrodes. More electrodes do not typically provide improved hearing performance. Given that this limitation is primarily due to current spread, and that newly developing kinds of electrodes may enable more focused stimulation, we recently proposed an information theoretic modeling framework for estimating how many electrodes might achieve optimal hearing performance under a range of assumptions about electrodes and their placement relative to the nerve. Here, we extend this approach by introducing more realistic three-dimensional spiral geometries for the cochlea and array, and comparing the optimal number of electrodes predicted by our model for this case with that in our original model, which used a linear geometry.

Optimized single pulse stimulation strategy for retinal implants

Retinal implants offer prospects of vision restoration for some blind patients by eliciting visual percepts of spots of light called 'phosphenes'. Recently, a mathematical model has been developed that predicts patients' perception of phosphene brightness for current-driven electrical stimulation of the retina. This model is explored for different stimulation parameters on a single electrode, including safety and hardware limitations, to produce phosphenes of specified brightness. We describe a procedure to derive stimulation parameters to account for such constraints, and describe methods to construct optimal stimuli in terms of producing maximal perceived brightness and efficient generation of phosphenes of a given brightness by employing minimal energy. In both cases, it is found that the resulting optimized stimulation waveforms consist of a long stimulation period, and interphase delays between initial and charge-balancing phases.

Modeling extracellular electrical stimulation: II. Computational validation and numerical results

The validity of approximate equations describing the membrane potential under extracellular electrical stimulation (Meffin et al 2012 J. Neural Eng. 9 065005) is investigated through finite element analysis in this paper. To this end, the finite element method is used to simulate a cylindrical neurite under extracellular stimulation. Laplace's equations with appropriate boundary conditions are solved numerically in three dimensions and the results are compared to the approximate analytic solutions. Simulation results are in agreement with the approximate analytic expressions for longitudinal and transverse modes of stimulation. The range of validity of the equations describing the membrane potential for different values of stimulation and neurite parameters are presented as well. The results indicate that the analytic approach can be used to model extracellular electrical stimulation for realistic physiological parameters with a high level of accuracy.

Retinal prosthesis safety: alterations in microglia morphology due to thermal damage and retinal implant contact

PURPOSE

In order to develop retinal implants with a large number of electrodes, it is necessary to ensure that they do not cause damage to the neural tissue by the heat that the electrical circuits generate. Knowledge about the threshold of the amount of power that induces damage will assist in developing power budgets for retinal implants.

METHODS

Heat-induced retinal damage was evaluated by measuring changes in the morphology of the resident immune cells, the microglia, which are the first cells to respond to retinal trauma. Microglial soma and arbor areas were assessed in rat retinal tissues in vitro to determine the effects of increasing temperatures, implant contact, and heating and implant contact combined.

RESULTS

In response to increasing incubation temperatures (no implant), microglial somas enlarged and arbor areas retracted, indicative of retinal stress. Thermal damage thresholds, defined as a significant change in microglial morphology from that observed at the upper limit of normal body temperature, occurred at a temperature of 38.7 °C. Implant contact, induced when a passive implant was placed on the retina, also caused similar morphological alterations in microglia, indicating retinal damage. Heated-implant contact exacerbated the effects of temperature alone but still resulted in a thermal damage threshold of 38.7 °C, the same as with heating alone.

CONCLUSIONS

Our conservative recommendations are that implanted retinal electronics keep power dissipations to less than 19 mW/mm(2) to stay below the microglial thermal damage threshold (2.1 °C) and to comply with international standards for implantable devices (2 °C).

Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations

Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace's equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite's membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.

An investigation of dendritic delay in octopus cells of the mammalian cochlear nucleus

Octopus cells, located in the mammalian auditory brainstem, receive their excitatory synaptic input exclusively from auditory nerve fibers (ANFs). They respond with accurately timed spikes but are broadly tuned for sound frequency. Since the representation of information in the auditory nerve is well understood, it is possible to pose a number of questions about the relationship between the intrinsic electrophysiology, dendritic morphology, synaptic connectivity, and the ultimate functional role of octopus cells in the brainstem. This study employed a multi-compartmental Hodgkin-Huxley model to determine whether dendritic delay in octopus cells improves synaptic input coincidence detection in octopus cells by compensating for the cochlear traveling wave delay. The propagation time of post-synaptic potentials from synapse to soma was investigated. We found that the total dendritic delay was approximately 0.275 ms. It was observed that low-threshold potassium channels in the dendrites reduce the amplitude dependence of the dendritic delay of post-synaptic potentials. As our hypothesis predicted, the model was most sensitive to acoustic onset events, such as the glottal pulses in speech when the synaptic inputs were arranged such that the model's dendritic delay compensated for the cochlear traveling wave delay across the ANFs. The range of sound frequency input from ANFs was also investigated. The results suggested that input to octopus cells is dominated by high frequency ANFs.

Wireless power delivery for retinal prostheses

Delivering power to an implanted device located deep inside the body is not trivial. This problem is made more challenging if the implanted device is in constant motion. This paper describes two methods of transferring power wirelessly by means of magnetic induction coupling. In the first method, a pair of transmit and receive coils is used for power transfer over a large distance (compared to their diameter). In the second method, an intermediate pair of coils is inserted in between transmit and receive coils. Comparison between the power transfer efficiency with and without the intermediate coils shows power transfer efficiency to be 11.5 % and 8.8 %, respectively. The latter method is especially suitable for powering implanted devices in the eye due to immunity to movements of the eye and ease of surgery. Using this method, we have demonstrated wireless power delivery into an animal eye.

Predicting phosphene elicitation in patients with retinal implants: a mathematical study

Single pulse waveforms were considered in a recent model for phosphene elicitation in patients with a retinal prosthesis. Waveforms are constrained to charge-balanced stimuli consisting of a single cathodic and anodic pulse pair. Mathematical models of such stimuli have been constructed and presented based upon patient testimonials. In this work, we derive analytic expressions that may be employed to determine equibrightness levels for different waveforms. We provide an example calculation to show quantitative improvements in stimulation efficiency that are consistent with qualitative findings on waveform effects.

Heating of the eye by a retinal prosthesis: modeling, cadaver and in vivo study

In order to develop retinal implants with a large number of electrodes, it is necessary to ensure that they do not cause damage to the neural tissue by the heat that the electrical circuits generate. Knowledge of the amount of power that induces thermal damage will assist in development of power budgets for implants, which has a significant effect upon the design of the prostheses circuitry. In this study, temperatures were measured at multiple locations on the retina while the retina was heated in cadaver and in vivo preparations using a variety of prosthesis implantation sites. A finite element thermal model of the cat eye was also created and validated by the cadaver and in vivo tests, allowing for a much larger spectrum of thermal influences to be evaluated without additional animal experimentation. To ensure that retinal tissue temperatures are not increased by more than 2 °C, a 5 mm × 5 mm, suprachoroidally implanted heating element must not dissipate more than 135 mW (5.4 mW/mm (2)).

Learning a Sparse Code for Temporal Sequences Using STDP and Sequence Compression

A spiking neural network that learns temporal sequences is described. A sparse code in which individual neurons represent sequences and subsequences enables multiple sequences to be stored without interference. The network is founded on a model of sequence compression in the hippocampus that is robust to variation in sequence element duration and well suited to learn sequences through spike-timing dependent plasticity (STDP). Three additions to the sequence compression model underlie the sparse representation: synapses connecting the neurons of the network that are subject to STDP, a competitive plasticity rule so that neurons specialize to individual sequences, and neural depolarization after spiking so that neurons have a memory. The response to new sequence elements is determined by the neurons that have responded to the previous subsequence, according to the competitively learned synaptic connections. Numerical simulations show that the model can learn sets of intersecting sequences, presented with widely differing frequencies, with elements of varying duration.

Thermal heating of a retinal prosthesis: thermal model and in-vitro study

In order to develop retinal implants with a large number of electrodes, it is necessary to ensure that they do not cause damage to the neural tissue by the heat that the electrical circuits generate. Knowledge about the threshold of the amount of power that induces thermal damage will greatly assist in development of power budgets for implants, which has a significant effect upon the design of implant circuitry. In this study, we developed and tested in-vitro equipment that can dissipate thermal energy in current prosthesis implantation sites while simultaneously measuring and recording temperature distributions at multiple locations along the retinal tissue. A finite element thermal model of the feline eye was also created and validated by the in-vitro tests allowing for a much larger spectrum of thermal influences to be evaluated without the additional cost of animal sacrifice.

Diamond penetrating electrode array for epi-retinal prosthesis

This paper presents progress in the characterization and application of diamond penetrating electrode arrays for Epi-Retinal Prostheses. Electrical stimulation of degenerate retina has already been shown to restore partial vision for some blind patients, albeit at low spatial resolution. Higher resolution may be achievable by building arrays with electrodes that have greater areal density and closer proximity to target neurons. However, high standards of biocompatibility and hermeticity must be maintained, limiting the range of available materials of manufacture. Here, the design and histology of high density electrode arrays (approximately 100 electrodes/mm(2)) made from polycrystalline diamond and implanted into rat retinae are discussed. Results from initial steps in this process are reported.

Differential stimulation of ON and OFF retinal ganglion cells: a modeling study

A model of the electrophysiological properties of ON and OFF retinal ganglion cells (RGCs) was constrained and validated using experimental data from the literature. Our simulations support experimental findings that differences in the magnitude of the T-type Ca(2+) current explain differences in the intrinsic electrophysiology of ON and OFF RGCs. The models are used to investigate the potential for differential stimulation of ON and OFF RGCs during neuroprosthetic stimulation with sinusoidal current. The model predicts that OFF cells fire preferential over ON cells in a frequency band around 10 Hz.

Theoretical framework for estimating the conductivity map of the retina through finite element analysis

A mathematical framework for estimation of the conductivity map of the retina is presented. The problem is formulated and solved in two-dimensional space considering hypothetical inhomogeneity in the conductivity profile at each layer of the retina in x and y directions. Finite element analysis is used to solve the equation of continuity in steady state to simulate voltage measurements as well as estimate the conductivity map. The results of simulated noisy data for an inhomogeneous retina layer and the fovea, which has a more complicated geometry, are presented. The error study of the estimated conductivity map shows that the error for an inhomogeneous conductivity profile is approximately 2% and the error for calculating the fovea conductivity map is just above 8%. This method can be extended to three-dimensions and can also be used to measure the impedance of different layers of the retina for alternating currents.

Simulating electrical stimulation of degenerative retinal ganglion cells with bi-phasic pulse trains

The aim of this work was to investigate how retinal ganglion cells (RGCs) respond to repetitive electrical stimulation in degenerative retina. The response of modeled ON and OFF cells was examined to bi-phasic pulse train stimulation of varying frequencies. Previously developed models of RGCs were extended to include an experimentally observable balance of excitatory and inhibitory currents in degenerative retina. The phenomena of fading and dark phosphenes with retinal implants were investigated. A hypothesis for a mechanism contributing to these phenomena was formulated.

Viability of the inner retina in a novel mouse model of retinitis pigmentosa

Retinal prostheses aim to restore vision to patients who are blind from photoreceptor diseases such as Retinitis Pigmentosa (RP). All implants target the neural cells in the inner retina, the retinal ganglion cells (RGCs). Our research focuses on further understanding the disease process of RP during mid to late stages when total loss of photoreceptors has occurred and significant remodeling of inner retinal neurons has taken place. We have used a novel transgenic mouse, Rd1-FTL, to observe different degenerative stages of RP. Notably, in the aged retina we have evidence that there was gross inner retinal remodeling as well as glial dysfunction that occurred in confined regions in the central retina that worsened overtime. Consequently, the timing of implantation and location of the prosthesis both need to account for the state of the retina at different stages in the disease process.

Spiking Neuron Model for Temporal Sequence Recognition

A biologically inspired neuronal network that stores and recognizes temporal sequences of symbols is described. Each symbol is represented by excitatory input to distinct groups of neurons (symbol pools). Unambiguous storage of multiple sequences with common subsequences is ensured by partitioning each symbol pool into subpools that respond only when the current symbol has been preceded by a particular sequence of symbols. We describe synaptic structure and neural dynamics that permit the selective activation of subpools by the correct sequence. Symbols may have varying durations of the order of hundreds of milliseconds. Physiologically plausible plasticity mechanisms operate on a time scale of tens of milliseconds; an interaction of the excitatory input with periodic global inhibition bridges this gap so that neural events representing successive symbols occur on this much faster timescale. The network is shown to store multiple overlapping sequences of events. It is robust to variation in symbol duration, it is scalable, and its performance degrades gracefully with perturbation of its parameters.

Selective filtering to spurious localization cues in the mammalian auditory brainstem

The cues used by mammals to localize sound can become corrupted when multiple sound sources are present due to the interference of sound waves. Under such circumstances these localization cues become spurious and often fluctuate rapidly (>100 Hz). By contrast, rapid fluctuations in sound pressure level do not indicate a corrupted signal, but rather may convey important information about the sound source. It is proposed that filtering in the auditory brainstem acts to selectively attenuate signals associated with the presence of rapidly fluctuating (spurious) localization cues, but not those associated with slowly varying cues. Further it is proposed that specific inhibitory circuitry in the auditory brainstem, centered on the dorsal nucleus of the lateral lemniscus (DNLL), contributes to this selective filtering. Data from extra-cellular recordings in anesthetized Mongolian gerbils are presented to support these hypotheses for a subpopulation of DNLL neurons. These results provide new insights into how the mammalian auditory system processes information about multiple sound sources.

Learning the structure of correlated synaptic subgroups using stable and competitive spike-timing-dependent plasticity

Synaptic plasticity must be both competitive and stable if ongoing learning of the structure of neural inputs is to occur. In this paper, a wide class of spike-timing-dependent plasticity (STDP) models is identified that have both of these desirable properties in the case in which the input consists of subgroups of synapses that are correlated within the subgroup through the occurrence of simultaneous input spikes. The process of synaptic structure formation is studied, illustrating one particular class of these models. When the learning rate is small, multiple alternative synaptic structures are possible given the same inputs, with the outcome depending on the initial weight configuration. For large learning rates, the synaptic structure does not stabilize, resulting in neurons without consistent response properties. For learning rates in between, a unique and stable synaptic structure typically forms. When this synaptic structure exhibits a bimodal distribution, the neuron will respond selectively to one or more of the subgroups. The robustness with which this selectivity develops during learning is largely determined by the ratio of the subgroup correlation strength to the number of subgroups. The fraction of potentiated subgroups is primarily determined by the balance between potentiation and depression.

Spike-Timing-Dependent Plasticity: The Relationship to Rate-Based Learning for Models with Weight Dynamics Determined by a Stable Fixed Point

Experimental evidence indicates that synaptic modification depends on the timing relationship between the presynaptic inputs and the output spikes that they generate. In this letter, results are presented for models of spike-timing-dependent plasticity (STDP) whose weight dynamics is determined by a stable fixed point. Four classes of STDP are identified on the basis of the time extent of their input-output interactions. The effect on the potentiation of synapses with different rates of input is investigated to elucidate the relationship of STDP with classical studies of long-term potentiation and depression and rate-based Hebbian learning. The selective potentiation of higher-rate synaptic inputs is found only for models where the time extent of the input-output interactions is input restricted (i.e., restricted to time domains delimited by adjacent synaptic inputs) and that have a time-asymmetric learning window with a longer time constant for depression than for potentiation. The analysis provides an account of learning dynamics determined by an input-selective stable fixed point. The effect of suppressive interspike interactions on STDP is also analyzed and shown to modify the synaptic dynamics.

An analytical model for the "large, fluctuating synaptic conductance state" typical of neocortical neurons in vivo

A model of in vivo-like neocortical activity is studied analytically in relation to experimental data and other models in order to understand the essential mechanisms underlying such activity. The model consists of a network of sparsely connected excitatory and inhibitory integrate-and-fire (IF) neurons with conductance-based synapses. It is shown that the model produces values for five quantities characterizing in vivo activity that are in agreement with both experimental ranges and a computer-simulated Hodgkin-Huxley model adapted from the literature (Destexhe et al. (2001) Neurosci. 107(1): 13-24). The analytical model builds on a study by Brunel (2000) (J. Comput. Neurosci. 8: 183-208), which used IF neurons with current-based synapses, and therefore does not account for the full range of experimental data. The present results suggest that the essential mechanism required to explain a range of data on in vivo neocortical activity is the conductance-based synapse and that the particular model of spike initiation used is not crucial. Thus the IF model with conductance-based synapses may provide a basis for the analytical study of the "large, fluctuating synaptic conductance state" typical of neocortical neurons in vivo.

Study of neuronal gain in a conductance-based leaky integrate-and-fire neuron model with balanced excitatory and inhibitory synaptic input

Neurons receive a continual stream of excitatory and inhibitory synaptic inputs. A conductance-based neuron model is used to investigate how the balanced component of this input modulates the amplitude of neuronal responses. The output spiking rate is well described by a formula involving three parameters: the mean mu and variance sigma of the membrane potential and the effective membrane time constant tauQ. This expression shows that, for sufficiently small tauQ, the level of balanced excitatory-inhibitory input has a nonlinear modulatory effect on the neuronal gain.