The somatic shunt cable model for neurons.

The derivation of the equations for an electrical model of nerve cells is presented. The model consists of an equivalent cylinder, a lumped somatic impedance, and a variable shunt at the soma. This shunt was introduced to take into account the fast voltage decays observed following the injections of current pulses in some motoneurons and hippocampal granule cells that could not be explained by existing models. The shunt can be interpreted either by penetration damage with the electrode or by a lower membrane specific resistance at the soma than in the dendrites. A solution of the model equations is presented that allows the estimation of the electrotonic length L, the membrane time constant tau m, the dendritic dominance ratio rho, and the shunt parameter epsilon, based only on the measurement of the first two coefficients and time constants in the multiexponential voltage response to injected current pulses.     

Identification of the cable parameters in the somatic shunt model

 We show that the first five moments of the soma potential and soma current uniquely and stably determine the soma conductance and capacitance and the dendritic electrotonic length, conductance, and capacitance in the so-called somatic shunt model of the passive behavior of a neuron. We test our resulting input admittance algorithm on synthetic data and demonstrate the regularizing effect of knowledge of the ratio of soma to dendrite surface areas. Received: 9 June 1999 / Accepted in revised form: 24 January 2000     

Techniques for obtaining analytical solutions to the multicylinder somatic shunt cable model for passive neurones.

The somatic shunt cable model for neurones is extended to the case in which several equivalent cylinders, not necessarily of the same electrotonic length, emanate from the cell soma. The cable equation is assumed to hold in each cylinder and is solved with sealed end conditions and a lumped soma boundary condition at a common origin. A Green's function (G) is defined, corresponding to the voltage response to an instantaneous current pulse at an arbitrary point along one of the cylinders. An eigenfunction expansion for G is obtained where the coefficients are determined using the calculus of residues and compared with an alternative method of derivation using a modified orthogonality condition. This expansion converges quickly for large time, but, for small time, a more convenient alternative expansion is obtained by Laplace transforms. The voltage response to arbitrary currents injected at arbitrary sites in the dendritic tree (including the soma) may then be expressed as a convolution integral involving G. Illustrative examples are presented for a point charge input.     

Analytical solutions to the multicylinder somatic shunt cable model for passive neurones with differing dendritic electrical parameters

The multicylinder somatic shunt cable model for passive neurones with differing time constants in each cylinder is considered in this paper. The solution to the model with general inputs is developed, and the parametric dependence of the voltage response is investigated. The method of analysis is straightforward and follows that laid out in Evans et al. (1992, 1994): (i) The dimensional problem is stated with general boundary and initial conditions, (ii) The model is fully non-dimensionalised, and a dimensionless parameter family which uniquely governs the behaviour of the dimensionless voltage response is obtained, (iii) The fundamental unit impulse problem is solved, and the solutions to problems involving general inputs are written in terms of the unit impulse solution, (iv) The large and small time behaviour of the unit impulse solution is examined, (v) The parametric dependence of the unit impulse upon the dimensionless parameter family is explored for two limits of practical interest. A simple expression for the principle relationship between the dimensionless parameter family is derived and provides insight into the interaction between soma and cylinders. A well-posed method for the solution of the dimensional inverse problem is presented.     

The parameter identification problem for the somatic shunt model

The somatic shunt model, a generalized version of the Rall equivalent cylinder model, is used commonly to describe the passive electrotonic properties of neurons. Procedures for determining the parameters of the somatic shunt model that best describe a given neuron typically rely on the response of the cell to a small step of hyperpolarizing current injected by an intrasomatic recording electrode. In this study it is shown that the problem of estimating model parameters for the somatic shunt model using physiological data is ill-posed, in that very small errors in measured data can lead to large and unpredictable errors in parameter estimates. If the somatic shunt is assumed to be a real property of the intact neuron, the effects of these errors are not severe when predicting EPSP waveshapes resulting from synaptic input at a given location. However, if the somatic shunt is assumed to be a consequence of a leakage pathway around the recording electrode, and a correction for the shunt is applied, then the instability of the inverse problem can introduce large errors in estimates of EPSP waveshape as a function of synaptic location in the intact cell. Morphological constraints can be used to improve the accuracy of the inversion procedure in terms of both parameter estimates and predicted EPSP responses.     

Balance between four types of synaptic input for the integrate-and-fire model

We consider the integrate-and-fire model with AMPA, NMDA, GABA(A)and GABA(B)synaptic inputs, with model parameters based upon experimental data. An analytical approach is presented to determine when a post-synaptic balance between excitation and inhibition can be achieved. Secondly, we compare the model behaviour subject to these four types of input, with its behaviour subject to conventional point process inputs. We conclude that point processes are not a good approximation, even away from exact presynaptic balance. Thirdly, numerical simulations are presented which demonstrate that we can treat NMDA and GABA(B)as DC currents. Finally, we conclude that a balanced input is plausible neither pre-synaptically nor post-synaptically for the model and parameters we employed.     

A cable model for coupled neurons with somatic gap junctions

A cable model is presented for a pair of electrotonically coupled neurons to investigate the spatial effects of soma-somatic gap junctions. The model extends that of Poznanski et al.(1995) in which each neuron is represented by a tapered equivalent cable attached to an isopotential soma with the two somas being electrically coupled. The model is posed generally, so that both active and passive properties can be considered. In the active case a system of nonlinear integral equations is derived for the voltage, whilst in the passive case these have an exact solution that also holds for inputs modelled as synaptic reversal potentials. Analytical and numerical methods are used to examine the sensitivity of the soma potentials (in particular) to the coupling resistance.     

The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.     

Transient synaptic ribbons in the mammalian retina at unusual sites

In the vertebrate retina the presence of synaptic ribbons (SRs) is well documented in two sites only, viz., in photoreceptor axon terminals in the outer plexiform layer and in bipolar cell axons in the inner plexiform layer. The present paper reports the presence of non-photoreceptor SRs in the outer plexiform layer of cattle and mouse, where they were seen in small numbers in thin cell processes near cone pedicles of light-adapted animals. They were never seen near rod spherules. Quantitative data obtained in mice killed at different time-points revealed that the SRs under consideration increased in number during day time and were absent during the dark phase. Moreover, under high light intensity of 10000 lux they were more frequent in number compared to 100-lux-exposed animals. It is concluded that the cell processes revealing the temporary presence of SRs are processes of flat bipolar cells which may provide a feedback to cones during the light phase.     

A general diffusion model for analyzing the efficacy of synaptic input to threshold neurons

We describe a general diffusion model for analyzing the efficacy of individual synaptic inputs to threshold neurons. A formal expression is obtained for the system propagator which, when given an arbitrary initial state for the cell, yields the conditional probability distribution for the state at all later times. The propagator for a cell with a finite threshold is written as a series expansion, such that each term in the series depends only on the infinite threshold propagator, which in the diffusion limit reduces to a Gaussian form. This procedure admits a graphical representation in terms of an infinite sequence of diagrams. To connect the theory to experiment, we construct an analytical expression for the primary correlation kernel (PCK) which profiles the change in the instantaneous firing rate produced by a single postsynaptic potential (PSP). Explicit solutions are obtained in the diffusion limit to first order in perturbation theory. Our approximate expression resembles the PCK obtained by computer simulation, with the accuracy depending strongly on the mode of firing. The theory is most accurate when the synaptic input drives the membrane potential to a mean level more than one standard deviation below the firing threshold, making such cells highly sensitive to synchronous synaptic input.     

The response of a model neurone to a white noise input

We consider a simple model of a neurone in which the input voltage is integrated to form the somatic potential, and a pulse is emitted when this reaches a threshold; the somatic potential is then reset to its resting value. We subject this model to a white-noise input, and evaluate the cross correlation between input white noise and the output pulse train; this is proportional to the small-signal impulse response of the model. Some numerical estimations are presented.     

Fusion of isolated synaptic vesicles as a model of the terminal stage of regulated exocytosis

In the study of membrane fusion, which is the terminal stage of exocytosis, we used a simplified model consisting of homotypic membranes of isolated synaptic vesicles (SV) obtained from the synaptosomal fraction of rat brain tissue. It was shown that fusion of SV develops in the presence of cytoplasmic proteins and 10^–7 to 10^–5 M Ca²⁺ ions. This conclusion was made based on changes in the intensity of fluorescence of a probe, R18. Calcium ions were found to be the most effective activators of the membrane fusion when the effects of bivalent cations, Ca²⁺, Sr²⁺, and Ba²⁺, were compared. ATP induced membrane fusion both in the presence and in the absence of Ca²⁺, and the effects of ATP and Ca²⁺ were additive. These findings allow us to believe that there are factors in the system containing SV and soluble proteins of synaptosomes, which initiate fusion of the membranes under the influence of not only Ca²⁺ but also ATP. The intensity of Ca²⁺-dependent fusion of SV dropped after trypsin treatment, i.e., proteolysis resulted in modulation of the sensitivity of vesicular proteins and/or a change in their capability of evoking membrane fusion. Monoclonal antibodies against synaptotagmin and synaptobrevin inhibited fusion of SV, but only partly. Our results support the concept that Ca²⁺-regulated membrane fusion is possible without the involvement of the entire SNARE complex.Neirofiziologiya/Neurophysiology, Vol. 36, No. 4, pp. 272–280, July–August, 2004.This revised version was published online in April 2005 with a corrected cover date.     

The role of additive neurogenesis and synaptic plasticity in a hippocampal memory model with grid-cell like input

Recently, we presented a study of adult neurogenesis in a simplified hippocampal memory model. The network was required to encode and decode memory patterns despite changing input statistics. We showed that additive neurogenesis was a more effective adaptation strategy compared to neuronal turnover and conventional synaptic plasticity as it allowed the network to respond to changes in the input statistics while preserving representations of earlier environments. Here we extend our model to include realistic, spatially driven input firing patterns in the form of grid cells in the entorhinal cortex. We compare network performance across a sequence of spatial environments using three distinct adaptation strategies: conventional synaptic plasticity, where the network is of fixed size but the connectivity is plastic; neuronal turnover, where the network is of fixed size but units in the network may die and be replaced; and additive neurogenesis, where the network starts out with fewer initial units but grows over time. We confirm that additive neurogenesis is a superior adaptation strategy when using realistic, spatially structured input patterns. We then show that a more biologically plausible neurogenesis rule that incorporates cell death and enhanced plasticity of new granule cells has an overall performance significantly better than any one of the three individual strategies operating alone. This adaptation rule can be tailored to maximise performance of the network when operating as either a short- or long-term memory store. We also examine the time course of adult neurogenesis over the lifetime of an animal raised under different hypothetical rearing conditions. These growth profiles have several distinct features that form a theoretical prediction that could be tested experimentally. Finally, we show that place cells can emerge and refine in a realistic manner in our model as a direct result of the sparsification performed by the dentate gyrus layer.     

A dendritic cable model for the amplification of synaptic potentials by an ensemble average of persistent sodium channels

The persistent sodium current density (I(NaP)) at the soma measured with the 'whole-cell' patch-clamp recording method is linearized about the resting state and used as a current source along the dendritic cable (depicting the spatial distribution of voltage-dependent persistent sodium ionic channels). This procedure allows time-dependent analytical solutions to be obtained for the membrane depolarization. Computer simulated response to a dendritic current injection in the form of synaptically-induced voltage change located at a distance from the recording site in a cable with unequally distributed persistent sodium ion channel densities per unit length of cable (the so-called 'hot-spots') is used to obtain conclusions on the density and distribution of persistent sodium ion channels. It is shown that the excitatory postsynaptic potentials (EPSPs) are amplified if hot-spots of persistent sodium ion channels are spatially distributed along the dendritic cable, with the local density of I(NaP) with respect to the recording site shown to specifically increase the peak amplitude of the EPSP for a proximally placed synaptic input, while the spatial distribution of I(NaP) serves to broaden the time course of the amplified EPSP. However, in the case of a distally positioned synaptic input, both local and nonlocal densities yield an approximately identical enhancement of EPSPs in contradiction to the computer simulations performed by Lipowsky et al. [J. Neurophysiol. 76 (1996) 2181]. The results indicate that persistent sodium channels produce EPSP amplification even when their distribution is relatively sparse (i.e. , approximately 1-2% of the transient sodium channels are found in dendrites of CA1 hippocampal pyramidal neurons). This gives a strong impetus for the use of the theory as a novel approach in the investigation of synaptic integration of signals in active dendrites represented as ionic cables.     

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 and variance of the membrane potential and the effective membrane time constant _Q. This expression shows that, for sufficiently small _Q, the level of balanced excitatory-inhibitory input has a nonlinear modulatory effect on the neuronal gain.     

Botulinum neurotoxin A blocks synaptic vesicle exocytosis but not endocytosis at the nerve terminal

The supply of synaptic vesicles in the nerve terminal is maintained by a temporally linked balance of exo- and endocytosis. Tetanus and botulinum neurotoxins block neurotransmitter release by the enzymatic cleavage of proteins identified as critical for synaptic vesicle exocytosis. We show here that botulinum neurotoxin A is unique in that the toxin-induced block in exocytosis does not arrest vesicle membrane endocytosis. In the murine spinal cord, cell cultures exposed to botulinum neurotoxin A, neither K(+)-evoked neurotransmitter release nor synaptic currents can be detected, twice the ordinary number of synaptic vesicles are docked at the synaptic active zone, and its protein substrate is cleaved, which is similar to observations with tetanus and other botulinal neurotoxins. In marked contrast, K(+) depolarization, in the presence of Ca(2+), triggers the endocytosis of the vesicle membrane in botulinum neurotoxin A-blocked cultures as evidenced by FM1-43 staining of synaptic terminals and uptake of HRP into synaptic vesicles. These experiments are the first demonstration that botulinum neurotoxin A uncouples vesicle exo- from endocytosis, and provide evidence that Ca(2+) is required for synaptic vesicle membrane retrieval.     

Transient corticosteroid treatment permanently amplifies the Th2 response in a murine model of asthma

Corticosteroids (CS) remain the most efficacious pharmacotherapeutic option for the management of asthma. Although the acute anti-inflammatory effects of CS treatment have been amply documented both clinically and experimentally, recent human data intimate that exposure to CS may be associated with retrograde immune phenomena, including enhanced synthesis of IgE in vivo and elevated Th2 cytokine production in vitro. We have investigated the long-term immunologic effects of CS treatment in a murine model of allergic airway inflammation. CS treatment during initial exposure to OVA or upon long-term Ag rechallenge remarkably attenuated eosinophilic airway inflammation and airway hyperresponsiveness. Interestingly, however, Th2 cytokine production by cultured splenocytes from CS-treated mice was significantly elevated, while IFN-gamma synthesis was depressed. Moreover, mice rechallenged with OVA several weeks after CS intervention during allergic sensitization not only developed airway inflammation, but also exhibited enhanced Th2 cytokine production in lymphoid tissues and OVA-specific IgE in serum. This amplification of the systemic immune response was associated with an intact APC compartment during CS-conditioned sensitization to OVA. These data indicate that immune processes underlying the allergic phenotype remain impervious to CS treatment and raise the possibility that treatment with CS during sensitization may amplify elements of the allergen-specific immune response.     

A model of photoreceptor cell for response to light through synaptic connection

A model is proposed for the responses of vertebrate photoreceptor cell to light stimuli. It is based on the findings that the resistance of visual cell membrane increases during illumination. In this model the relation between the changes of membrane resistance and light intensity through synaptic connection is considered. This model suggests the general relation between the peak amplitude of receptor response and the intensity of flash.