Computer software for development and simulation of compartmental models of neurons

Is realistic neuronal modeling realistic?

Scientific models are abstractions that aim to explain natural phenomena. A successful model shows how a complex phenomenon arises from relatively simple principles while preserving major physical or biological rules and predicting novel experiments. A model should not be a facsimile of reality; it is an aid for understanding it. Contrary to this basic premise, with the 21st century has come a surge in computational efforts to model biological processes in great detail. Here we discuss the oxymoronic, realistic modeling of single neurons. This rapidly advancing field is driven by the discovery that some neurons don't merely sum their inputs and fire if the sum exceeds some threshold. Thus researchers have asked what are the computational abilities of single neurons and attempted to give answers using realistic models. We briefly review the state of the art of compartmental modeling highlighting recent progress and intrinsic flaws. We then attempt to address two fundamental questions. Practically, can we realistically model single neurons? Philosophically, should we realistically model single neurons? We use layer 5 neocortical pyramidal neurons as a test case to examine these issues. We subject three publically available models of layer 5 pyramidal neurons to three simple computational challenges. Based on their performance and a partial survey of published models, we conclude that current compartmental models are ad hoc, unrealistic models functioning poorly once they are stretched beyond the specific problems for which they were designed. We then attempt to plot possible paths for generating realistic single neuron models.

Accelerating compartmental modeling on a graphical processing unit

Compartmental modeling is a widely used tool in neurophysiology but the detail and scope of such models is frequently limited by lack of computational resources. Here we implement compartmental modeling on low cost Graphical Processing Units (GPUs), which significantly increases simulation speed compared to NEURON. Testing two methods for solving the current diffusion equation system revealed which method is more useful for specific neuron morphologies. Regions of applicability were investigated using a range of simulations from a single membrane potential trace simulated in a simple fork morphology to multiple traces on multiple realistic cells. A runtime peak 150-fold faster than the CPU was achieved. This application can be used for statistical analysis and data fitting optimizations of compartmental models and may be used for simultaneously simulating large populations of neurons. Since GPUs are forging ahead and proving to be more cost-effective than CPUs, this may significantly decrease the cost of computation power and open new computational possibilities for laboratories with limited budgets.

Trends in programming languages for neuroscience simulations

Neuroscience simulators allow scientists to express models in terms of biological concepts, without having to concern themselves with low-level computational details of their implementation. The expressiveness, power and ease-of-use of the simulator interface is critical in efficiently and accurately translating ideas into a working simulation. We review long-term trends in the development of programmable simulator interfaces, and examine the benefits of moving from proprietary, domain-specific languages to modern dynamic general-purpose languages, in particular Python, which provide neuroscientists with an interactive and expressive simulation development environment and easy access to state-of-the-art general-purpose tools for scientific computing.

Whole-cell voltage clamp recording

This unit describes the use of whole-cell voltage clamping to study voltage-gated channels. Stepwise changes in voltage produced by this technique cause channels to interconvert between different states, and these transitions are monitored as changes in membrane current. The time course of this redistribution of states contains a great deal of information about the mechanism of channel gating. Furthermore, the voltage clamp can be used to activate different populations of channels selectively. In this way, a specific channel targeted by biological or pharmacological manipulations can often be identified and studied in detail. This technique is also readily adapted to the study of ligand-gated channels, synaptic potentials, and exocytosis.

A software for simulating steady-state properties of passive dendrites based on the cable theory

In this study, a computer software, CableTeo, is introduced for simulating the steady-state electrical properties of passive dendrite based on the cable theory. The cable theory for dendritic neurons addresses to current-voltage relations in a continuous passive dendritic tree. It is briefly summarized that the cable theory related to passive cables and dendrites, which is a useful approximation and an important reference for excitable cases. The proposed software can be used to construct user-defined dendritic tree model. The user can define the model in detail, display the constructed dendritic tree, and examine the basic electrical properties of the dendritic tree using transfer impedance approach. The software addresses to ones who want to run simple simulations of the cable theory without need to any programming language skills or expensive software. It can also be used for educational purposes.

A computer software for simulating single-compartmental model of neurons

In this paper, a new computer software package, Yalzer, is introduced for simulating single-compartmental model of neurons. Passive or excitable membranes with voltage-gated ion channels can be modeled, and current clamp and voltage clamp experiments can be simulated. In the Yalzer, first-order differential equations used to define the dynamics of the gate variables and the membrane potential are solved by two separate integration methods with variable time steps: forward Euler and exponential Euler methods. Outputs of the simulation are shown on a spreadsheet template for allowing flexible data manipulation and can be graphically displayed. The user can define the model in detail, and examine the excitability of the model and the dynamics of voltage-gated ion channels. The software package addresses to ones who want to run simple simulations of neurons without need to any programming language skills or expensive software. It can also be used for educational purposes.

Pituitary control of BK potassium channel function and intrinsic firing properties of adrenal chromaffin cells

The discovery that the hypothalamic-pituitary-adrenocortical (HPA) endocrine stress axis controls an alternative splicing decision in chromaffin Slo-encoded BK (big potassium) channels raised the possibility that activation of the HPA could serve as a mechanism to tune the intrinsic electrical properties of epinephrine-secreting adrenal chromaffin cells. To test this, we compared BK functional properties and cell excitability in chromaffin cells from normal and hypophysectomized (pituitary-ablated) rats. Hypophysectomy was found to alter the voltage dependence and kinetics of BK gating, making channels less accessible for activation from rest. Perforated-patch recordings revealed changes in action potential waveform and repetitive firing properties. The maximum number of spikes that could be elicited with a 2 sec depolarizing current pulse was reduced by approximately 50% by hypophysectomy. The results indicate that pituitary hormones can adapt the mechanics of adrenal catecholamine release by tailoring BK channel function.

A simple stochastic model of spatially complex neurons

A method for studying the coding properties of a multicompartmental integrate-and-fire neuron of arbitrary geometry is presented. Depolarization at each compartment evolves like a leaky integrator with an after-firing reset imposed only at the trigger zone. The frequency of firing at the steady-state regime is related to the properties of the multidimensional input. The decreasing variability of subthreshold depolarization from the dendritic tree to the trigger zone is shown for an input that is corrupted by a white noise. The role of a Poissonian noise is also investigated. The proposed method gives an estimate of the mean interspike interval that can be used to study the input output transfer function of the system. Both types of the stochastic inputs result in broadening the transfer function with respect to the deterministic case.

A comparison of human motoneuron data to simulated data using cat motoneuron models

The response of repetitively firing human motoneurons to a composite excitatory input was evaluated. It was clearly shown that the response of the motoneurons to the transient input decreased with an increase in the background firing rate of the cell. The current model of repetitively firing human motoneurons could not account for this experimental result. Therefore, a compartmental modelling approach was used to simulate the repetitive firing properties of anaesthetised cat motoneurons under current clamp conditions. The modelled motoneurons were used in simulations similar to the experimental paradigms where the response to a composite excitatory input was evaluated at different background firing rates. The motoneuron models also showed a decrease in response to the excitatory input at faster background firing rates. The results suggest that human motoneurons are more comparable to motoneurons in the anaesthetised cat preparation than formerly thought. The results also demonstrate that the apparent efficacy of a synaptic input may be modulated by changes in background firing rate of the postsynaptic neuron.

Distal versus proximal inhibitory shaping of feedback excitation in the electrosensory lateral line lobe: implications for sensory filtering

Distal versus proximal inhibitory shaping of feedback excitation in the electrosensory lateral line lobe: implications for sensory filtering. J. Neurophysiol. 80: 3214-3232, 1998. The inhibition controlling the indirect descending feedback (parallel fibers originating from cerebellar granule cells in the eminentia posterior pars granularis) to electrosensory lateral line lobe (ELL) pyramidal cells was studied using intracellular recording techniques in vitro. Parallel fibers (PF) contact stellate cells and dendrites of ventral molecular layer (VML) GABAergic interneurons. Stellate cells provide local input to pyramidal cell distal dendrites, whereas VML cells contact their somata and proximal dendrites. Single-pulse stimulation of PF evoked graded excitatory postsynaptic potentials (EPSPs) that were blocked by alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid and N-methyl--aspartate (NMDA) antagonists. The EPSPs peaked at 6.4 +/- 1.8 ms (mean +/- SE; n = 11) but took >50 ms to decay completely. Tetanic stimulation (100 ms, 100 Hz) produced a depolarizing wave with individual EPSPs superimposed. The absolute amplitude of the individual EPSPs decreased during the train. Spike rates, established by injected current, mostly were increased, but in some cells were decreased, by tetanic stimulation. Global application of a gamma-aminobutyric acid-A (GABAA) antagonist to the recorded cell's soma and apical dendritic region increased the EPSP peak and decay phase amplitudes. Tetanic stimulation always increased current-evoked spike rates after GABAA blockade during, and for several hundred milliseconds after, the stimulus. Application of a GABAB antagonist did not have any significant effects on the PF-evoked response. This, and the lack of any long hyperpolarizing inhibitory postsynaptic potentials, suggests that VML and stellate cell inhibition does not involve GABAB receptors. Focal GABAA antagonist applications to the dorsal molecular layer (DML) and pyramidal cell layer (PCL) had contrasting effects on PF-evoked EPSPs. DML GABAA blockade significantly increased the EPSP peak amplitude but not the decay phase of the EPSP, whereas PCL GABAA-blockade significantly increased the decay phase, but not the EPSP peak, amplitude. The order of antagonist application did not affect the outcome. On the basis of the known circuitry of the ELL, we conclude that the distal inhibition originated from GABAergic molecular layer stellate cells and the proximal inhibition originated from GABAergic cells of the ventral molecular layer (VML cells). Computer modeling of distal and proximal inhibition suggests that intrinsic differences in IPSP dynamics between the distal and proximal sites may be amplified by voltage-dependent NMDA receptor and persistent sodium currents. We propose that the different time courses of stellate cell and VML cell inhibition allows them to act as low- and high-pass filters respectively on indirect descending feedback to ELL pyramidal cells.

Inhibition evoked from primary afferents in the electrosensory lateral line lobe of the weakly electric fish (Apteronotus leptorhynchus)

Inhibition evoked from primary afferents in the electrosensory lateral line lobe of the weakly electric fish (Apteronotus leptorhynchus). J. Neurophysiol. 80: 3173-3196, 1998. The responses of two types of projection neurons of the electrosensory lateral line lobe, basilar (BP) and nonbasilar (NBP) pyramidal cells, to stimulation of primary electrosensory afferents were determined in the weakly electric fish, Apteronotus leptorhynchus. Using dyes to identify cell type, the response of NBP cells to stimulation of primary afferents was inhibitory, whereas the response of BP cells was excitation followed by inhibition. gamma-Aminobutyric acid (GABA) applications produced biphasic (depolarization then hyperpolarization) responses in most cells. GABAA antagonists blocked the depolarizing effect of GABA and reduced the hyperpolarizing effect. The GABAB antagonists weakly antagonized the hyperpolarizing effect. The early depolarization had a larger increase in cell conductance than the late hyperpolarization. The conductance changes were voltage dependent, increasing with depolarization. In both cell types, baclofen produced a slow small hyperpolarization and reduced the inhibitory postsynaptic potentials (IPSPs) evoked by primary afferent stimulation. Tetanic stimulation of primary afferents at physiological rates (100-200 Hz) produced strongly summating compound IPSPs (approximately 500-ms duration) in NBP cells, which were usually sensitive to GABAA but not GABAB antagonists; in some cells there remained a slow IPSP that was unaffected by GABAB antagonists. BP cells responded with excitatory or mixed excitatory + inhibitory responses. The inhibitory response had both a fast (approximately 30 ms, GABAA) and long-lasting slow phase (approximately 800 ms, mostly blocked by GABAA antagonists). In some cells there was a GABAA antagonist-insensitive slow IPSP (approximately 500 ms) that was sensitive to GABAB antagonists. Application of glutamate ionotropic receptor antagonists blocked the inhibitory response of NBP cells to primary afferent stimulation and the excitatory response of BP cells but enhanced the BP cell slow IPSP; this remaining slow IPSP was reduced by GABAB antagonists. Unit recordings in the granule cell layer and computer simulations of pyramidal cell inhibition suggested that the duration of the slow GABAA inhibition reflects the prolonged firing of GABAergic granule cell interneurons to primary afferent input. Correlation of the results with known GABAergic circuitry in the electrosensory lobe suggests that the GABAergic type 2 granule cell input to both pyramidal cell types is via GABAA receptors. The properties of the GC2 GABAA input are well suited to their putative role in gain control, regulation of phasicness, and coincidence detection. The slow GABAB IPSP evoked in BP cells is likely due to ovoid cell input to their basal dendrites.

Analysis of the signals generated by networks of neurons coupled to planar arrays of microtransducers in simulated experiments

Planar microelectrode arrays can be used to characterize the dynamics of networks of neurons reconstituted in vitro. In this paper simulations related to experiments of the electrical activity recording by means of planar arrays of microtransducers coupled to networks of neurons are described. First a detailed model of single and synaptically connected neurons is given, appropriate to computer simulate the action potentials of neuronal populations. Then 'realistic' signals are generated. These signals are intended to reproduce, both in shape and intensity, those recorded by a microelectrode array. Typical experimental conditions are considered, and a detailed analysis given, of the bioelectronic coupling and of its influence on the shape of the recorded signals. Finally, simulated experiments dealing with dorsal root ganglia neurons are described and analysed in comparison with experimental results reported in the literature and obtained in our own laboratory. The effectiveness of the planar microelectrode technique is briefly discussed.

A simple program for simulating the responses of neurons with arbitrarily structured and active dendritic trees

We describe a simple program to simulate neural responses. This program is based on the compartmental approach, in which all compartments of a neuron (i.e. axon, soma or dendrite) are represented by the same basic electrical structure. A parameter file is used to store the model parameters, including the nonlinear channel characteristics of each compartment. The model is then automatically configured according to the values specified in the parameter file. The computation of the conductance of each active channel over time is handled by a unique subroutine optimized according to the kinetics of each channel. The equations for arbitrarily structured trees are solved implicitly using a simple algorithm similar to that of Hines (Hines, M. (1984) Int. J. Biomed. Comput., 15:69-76). The output of the model uses PostScript format. The advantage of this program is that it is small in size, simple to use, efficient, and is platform independent.

Computer simulation of the responses of human motoneurons to composite 1A EPSPS: effects of background firing rate

Two compartmental models of spinal alpha motoneurons were constructed to explore the relationship between background firing rate and response to an excitatory input. The results of these simulations were compared with previous results obtained from human motoneurons and discussed in relation to the current model for repetitively firing human motoneurons. The morphologies and cable parameters of the models were based on two type-identified cat motoneurons previously reported in the literature. Each model included five voltage-dependent channels that were modeled using Hodgkin-Huxley formalism. These included fast Na+ and K+ channels in the initial segment and fast Na+ and K+ channels as well as a slow K+ channel in the soma compartment. The density and rate factors for the slow K+ channel were varied until the models could reproduce single spike AHP parameters for type-identified motoneurons in the cat. Excitatory synaptic conductances were distributed along the equivalent dendrites with the same density described for la synapses from muscle spindles to type-identified cat motoneurons. Simultaneous activation of all synapses on the dendrite resulted in a large compound excitatory postsynaptic potential (EPSP). Brief depolarizing pulses injected into a compartment of the equivalent dendrite resulted in pulse potentials (PPs), which resembled the compound EPSPs. The effects of compound EPSPs and PPs on firing probability of the two motoneuron models were examined during rhythmic firing. Peristimulus time histograms, constructed between the stimulus and the spikes of the model motoneuron, showed excitatory peaks whose integrated time course approximated the time course of the underlying EPSP or PP as has been shown in cat motoneurons. The excitatory peaks were quantified in terms of response probability, and the relationship between background firing rate and response probability was explored. As in real human motoneurons, the models exhibited an inverse relationship between response probability and background firing rate. The biophysical properties responsible for the relationship between response probability and firing rate included the shapes of the membrane voltage trajectories between spikes and nonlinear changes in PP amplitude during the interspike interval at different firing rates. The results from these simulations suggest that the relationship between response probability and background firing rate is an intrinsic feature of motoneurons. The similarity of the results from the models, which were based on the properties of cat motoneurons, and those from human motoneurons suggests that the biophysical properties governing rhythmic firing in human motoneurons are similar to those of the cat.

Activation of intrinsic and synaptic currents in leech heart interneurons by realistic waveforms

Leech heart interneurons were voltage-clamped with realistic waveforms to investigate the currents underlying the oscillation in the cells. By estimating the leak current parameters in regions in which there was little contamination by voltage-gated currents, it was possible to measure the Ca2+ current, the persistent Na+ current, Ip, and the hyperpolarization-activated inward current, Ih. The experiments verified a prediction of a computer model of HN cells that the shape of the typical waveform was such that the low-threshold Ca2+ currents were partially inactivated during a slow up-ramp to a plateau potential. A step within the same range of the membrane potential as the realistic waveform produced > 4 times as much Ca2+ current. In two-cell voltage-clamp experiments, the step produced 20 times more graded inhibition than the normal presynaptic waveform. When the presynaptic heart interneuron oscillated with spikes, the graded inhibition was larger. The difference may arise from integration of a slowly decaying component of the spike-mediated inhibition. The persistent Na+ current had a very low threshold. During the most hyperpolarized phase of the waveform, Ip deactivated to 50% of its maximum conductance. A substantial part of Ip, therefore, was effectively contributing to the leak current in the HN cells. The h-current increased for waveforms that had longer periods, whereas increasing the h-current in the model reduced the period. The h-current thus provides negative feedback to perturbations that alter the period of the oscillation.

Layer-specific properties of the transient K current (IA) in piriform cortex

Piriform cortex in the rat is highly susceptible to induction of epileptiform activity. Experiments in vivo and in vitro indicate that this activity originates in endopiriform nucleus (EN). In slices, EN neurons are more excitable than layer II (LII) pyramidal cells, with more positive resting potentials and lower spike thresholds. We investigated potassium currents in EN and LII to evaluate their contribution to these differences in excitability. Whole-cell currents were recorded from identified cells in brain slices. A rapidly inactivating outward current (IA) had distinct properties in LII (IA,LII) versus EN (IA,EN). The peak amplitude of IA,EN was 45% smaller than IA,LII, and the kinetics of activation and inactivation was significantly slower for IA,EN. The midpoint of steady-state inactivation was hyperpolarized by 10 mV for IA,EN versus IA,LII, whereas activation was similar in the two cell groups. Other voltage-dependent potassium currents were indistinguishable between EN and LII. Simulations using a compartmental model of LII cells argue that different cellular distributions of IA channels in EN versus LII cells cannot account for these differences. Thus, at least some of the differences are intrinsic to the channels themselves. Current-clamp simulations suggest that the differences between IA,LII and IA,EN can account for the observed difference in resting potentials between the two cell groups. Simulations show that this difference in resting potential leads to longer first spike latencies in response to depolarizing stimuli. Thus, these differences in the properties of IA could make EN more susceptible to induction and expression of epileptiform activity.

The impact of receptor desensitization on fast synaptic transmission

The role of desensitization of ligand-gated channels at fast chemical synapses has been difficult to establish. Densensitization has been studied traditionally with prolonged agonist exposure, whereas the duration of free neurotransmitter in the synaptic cleft is relatively brief. Studies of acetylcholine-, glutamate- and GABA-gated channels using rapid agonist application now provide a means to assess the effects of densensitization in shaping synaptic responses and in influencing neuronal excitability. These data reveal several strikingly different patterns by which the receptor-specific kinetics of densensitization can determine the size, timecourse and frequency of transmitted signals. Densensitization is thus a surprisingly versatile mechanism for shaping synaptic transmission.

Heartbeat control in the medicinal leech: a model system for understanding the origin, coordination, and modulation of rhythmic motor patterns

We have analyzed in detail the neuronal network that generates heartbeat in the leech. Reciprocally inhibitory pairs of heart interneurons form oscillators that pace the heartbeat rhythm. Other heart interneurons coordinate these oscillators. These coordinating interneurons, along with the oscillators interneurons, form an eight-cell timing oscillator network for heartbeat. Still other interneurons, along with the oscillator interneurons, inhibit heart motor neurons, sculpting their activity into rhythmic bursts. Critical switch interneurons interface between the oscillator interneurons and the other premotor interneurons to produce two alternating coordination states of the motor neurons. The periods of the oscillator interneurons are modulated by endogenous RFamide neuropeptides. We have explored the ionic currents and graded and spike-mediated synaptic transmission that promote oscillation in the oscillator interneurons and have incorporated these data into a conductance-based computer model. This model has been of considerable predictive value and has led to new insights into how reciprocally inhibitory neurons produce oscillation. We are now in a strong position to expand this model upward, to encompass the entire heartbeat network, horizontally, to elucidate the mechanisms of FMRFamide modulation, and downward, to incorporate cellular morphology. By studying the mechanisms of motor pattern formation in the leech, using modeling studies in conjunction with parallel physiological experiments, we can contribute to a deeper understanding of how rhythmic motor acts are generated, coordinated, modulated, and reconfigured at the level of networks, cells, ionic currents, and synapses.

Action potential propagation and propagation block by GABA in rat posterior pituitary nerve terminals

1. A theoretical model was developed to investigate action potential propagation in posterior pituitary nerve terminals. This model was then used to evaluate the efficacy of depolarizing and shunting GABA responses on action potential propagation. 2. Experimental data obtained from the posterior pituitary with patch clamp techniques were used to derive empirical expressions for the voltage and time dependence of the nerve terminal Na+ and K+ channels. The essential structure employed here was based on anatomical and cable data from the posterior pituitary, and consisted of a long cylindrical axon (diameter, 0.5 mm) with a large spherical swelling (diameter, 4-21 mm) in the middle. 3. In the absence of an inhibitory conductance, simulated action potentials propagated with high fidelity through the nerve terminal. Swellings could block propagation, but only when sizes exceeded those observed in the posterior pituitary. Adding axonal branches reduced the critical size only slightly. These results suggested that action potentials invade the entire posterior pituitary nerve terminal in the absence of inhibition or depression. 4. The addition of inhibitory conductance to a swelling caused simulated action potentials to fail at the swelling. Depolarizing inhibitory conductances were 1.6 times more effective than shunting inhibitory conductances in blocking propagation. 5. Inhibitory conductances within the range of experimentally observed magnitudes and localized to swellings in the observed range of sizes were too weak to block simulated action potentials. However, twofold enhancement of GABA responses by neurosteroid resulted in currents strong enough to block propagation in realistic swelling sizes. 6. GABA could block simulated propagation without neurosteroid enhancement provided that GABA was present throughout a region in the order of a few hundred micrometres. For this widespread inhibition depolarizing conductance was 2.2 times more effective than shunting conductance. 7. These results imply two modes of propagation block, one resulting from highly localized release of inhibitory transmitter under conditions potentiating GABA responses, and the other resulting from widespread release of GABA in the absence of receptor potentiation. 8. The Na+ channels of the posterior pituitary nerve terminal have a unique voltage dependence that allows small depolarizations to inactivate without causing activation. The voltage dependence of this Na+ channel may serve as a specialized adaptation that facilitates in allowing small depolarizing conductances to block action potential propagation.

New computational models of the respiratory oscillator in mammals

Understanding how respiratory oscillations are generated and controlled in the mammalian brainstem requires synthesis of neural function at cellular, synaptic, and network levels. Modeling the respiratory network has entered a new era where simulations incorporating complex cellular properties and network interactions exhibited by respiratory neurons are possible. Given the complexity and unknowns of the real system, there are obvious limitations of the modeling in its present form. Nevertheless, if we are to produce neurobiologically satisfying, mechanistic explanations that synthesize cellular and network phenomena, the modeling approach outlined here should have distinct advantages. The approach allows a close interaction between experimental and computational studies, and will enable simulations with increasing neurobiological realism as additional experimental data becomes available. The modeling summarized here represents the initial stages of establishing a computational framework for storing information about neuron and network properties, and for exploring complex dynamical properties in ways not possible experimentally.

Realistic simulations of neurons by means of an ad hoc modified version of SPICE

This paper describes an ad hoc modified version of the electrical circuit analysis program SPICE, which has been optimized for detailed simulations of the behaviour of neurons. An equivalent-circuit description of the simulation building blocks is provided, and the SPICE modifications are specified. These modifications, in contrast to previous uses of SPICE, allows one to simulate the behaviour of neurons of Hodgkin-Huxley type (excitable membrane) and of postsynaptic membranes without any approximations. Simulation results are reported and compared, both with data previously analysed in the literature by other authors and with experimental data recently obtained by coupling neurons to planar extracellular microelectrodes. Details of the circuit elements used in the simulations are reported. The improvements of our proposed model are discussed in comparison with a previous SPICE-based model described in the literature.

Motor-pattern-generating networks in invertebrates: modeling our way toward understanding

Motor-pattern-generating networks in invertebrates have been the objects of intensive study to determine the origin and modulation of rhythmic neural activity. In some pattern generators, intrinsically bursting neurons drive activity throughout the network. In most pattern generators, however, rhythmicity arises from the interplay between intrinsic membrane properties and synaptic interaction. Reciprocal inhibitory synapses between neurons are thought to be crucial for generating oscillation in these networks, but a fundamental understanding of how such network oscillators work remains elusive. Progress towards this goal has come from attempts to combine computational modeling approaches with conventional physiological analysis.

Propagation of action potentials along complex axonal trees. Model and implementation

Axonal trees are typically morphologically and physiologically complicated structures. Because of this complexity, axonal trees show a large repertoire of behavior: from transmission lines with delay, to frequency filtering devices in both temporal and spatial domains. Detailed theoretical exploration of the electrical behavior of realistically complex axonal trees is notably lacking, mainly because of the absence of a simple modeling tool. AXONTREE is an attempt to provide such a simulator. It is written in C for the SUN workstation and implements both a detailed compartmental modeling of Hodgkin and Huxley-like kinetics, and a more abstract, event-driven, modeling approach. The computing module of AXONTREE is introduced together with its input/output features. These features allow graphical construction of arbitrary trees directly on the computer screen, and superimposition of the results on the simulated structure. Several numerical improvements that increase the computational efficiency by a factor of 5-10 are presented; most notable is a novel method of dynamic lumping of the modeled tree into simpler representations ("equivalent cables"). AXONTREE's performance is examined using a reconstructed terminal of an axon from a Y cell in cat visual cortex. It is demonstrated that realistically complicated axonal trees can be handled efficiently. The application of AXONTREE for the study of propagation delays along axonal trees is presented in the companion paper (Manor et al., 1991).