Neural Systems: Analysis and Modeling

Most of us share the qualitative impression that the nervous system is a remarkable computational device. Part of the evidence for this belief is the tremendous difficulty in imitating neural functions using man-made hardware — we are a long way from having machines which see or hear. Yet it is easy to find tasks where even rather modest modern computers far outperform relatively sophisticated animals. What then is so special about neural computation?

All of an organism’s knowledge about the sensory world comes from observation of the spike trains in its own sensory cells. How much information is carried in these spike trains Nand how efficient is the coding? We have recently developed an approach to neural coding which allows us to measure the information a spike train carries about a sensory stimulus without assumptions about how the information is coded[1]Comparing this measured information with an upper bound to the information rate determined by the spiking statistics results in a direct measure of the efficiency of coding. For two different mechanoreceptors from the cricket cercal system and from the bullfrog sacculus we find information rates close to 3 bits per spike (corresponding to 300 bits per sec in the cricket 160 bits per sec in the frog) and coding efficiencies greater than 50%.

We present a technique for the analysis of pattern formation by a class of models for the formation of ocular dominance stripes in the striate cortex of some mammals. The method, which employs the adiabatic approximation to derive a set of ordinary differential equations for patterning modes, has been successfully applied to reaction-diffusion models for striped patterns [1]. Models of ocular dominance stripes have been studied [2,3] by computation, or by linearization of the model equations. These techniques do not provide a rationale for the origin of the stripes. We show here that stripe formation is a non-linear property of the models. Our analysis indicates that stripe selection is closely linked to a property in the dynamics of the models which arises from a symmetry between ipsilateral and contralateral synapses to the visual cortex of a given hemisphere.

We show how hierarchical networks may be constructed of interconnected oscillatory network modules developed previously as models of olfactory cortex, or caricatures of “patches”of neocortex. The architecture is such that the larger system is itself a special case of the type of network of the submodules, and can be analysed with the same tools used to design the subnetwork modules. A particular subnetwork is formed by a set of neural populations whose interconnections also contain higher order synapses. These synapses determine attractors for that subnetwork independent of other subnetworks. Each subnetwork module assumes only minimal coupling justified by known anatomy. An N node module can be shown to function as an associative memory for up to N/2 oscillatory and N/3 chaotic memory attractors.The modules can learn connection weights between themselves which will cause the system to evolve under a clocked “machine cycle” by a sequence of transitions of attractors within the modules, much as a digital computer evolves by transitions of its binary flip-flop states. Thus the architecture employs the principle of “computing with attractors” used by macroscopic systems for reliable computation in the presence of noise. Clocking is done by rhythmic variation of certain bifurcation parameters which hold sensory modules clamped at their attractors while motor states change, and then clamp motor states while sensory states are released to take new states based on input from external motor output and internal feedback.Simulations show robust storage of oscillatory attractor transition sequences in a system with a sinusoidal clock and continuous oscillatory intermodule driving. The phase-locking or “binding” which occurs rapidly between coupled attractors of similar resonant frequency in different modules is important for reliable transitions. An oscillating systm with phase-locking modules is more robust to perturbation by additive Gaussian noise than an otherwise identical system with static attractors (frequencies set to zero). We show analytically how modular networks with more fault tolerant and biologically plausible distributed patterns can be built from “spreading activation” style networks which use single node representations.

We have developed an integrate-and-fire spike-initiation model based on that of Hill (1936), which comprises a pair of coupled, linear differential equations describing membrane potential and a threshold variable. In our version, we add an intrinsic noise source and explicit representation of the spike and its effect on threshold. The equations of the Hill model describe the time evolution of membrane potential and threshold in the subthreshold region. The model has three time constants. We demonstrate that, with suitable choice of these time constants and the mean value of the intrinsic noise, the model can mimic the spike-response characteristics of many classes of neurons. Specifically, with respect to auditory brainstem neurons, we produce the responses characteristic of the primary-type neuron, several varieties of chopper cell and onset cells, all of which are found in the mammalian cochlear nucleus. We demonstrate that the model computes simulated spike responses quickly.

We have developed a simplified description of voltage dependent channels to describe the currents involved in spike generation. We included this spiking model in a compartmental neuron model in order to obtain the correct scaling factor between the current actually injected in an experiment and the current arriving at the spike initiating zone. After fixing the parameters for our spiking model to match the spike shape of the real neuron the model had a strong bifurcation in its intensity versus rate characteristic. The rate of firing would abruptly jump from 0 to a value close to the saturating rate as the intensity of the stimulus was increased. To eliminate this artificial singularity we looked at the effect of both an additional shunting current and of current noise as suggested by previous researchers. We found that either the shunt or the noise alone could get rid of the bifurcation but that only a specific combination of shunt and noise would also yield the correct spiking variability as seen in interspike histograms.

The use of numerical simulation techniques to study neurobiology has grown over the past few years to encompass systems ranging from molecular level processes occurring in real neurons to systems of many thousands of highly abstract neuron-like elements (Touretzky, 1990 Koch and Segev, 1989; Zornetzer et al 1990). It is increasingly apparent that all these levels of analysis are interrelated and may be essential to the understanding of the functioning of the nervous system (Bower, 1991)

We describe CAJAL-91 a programming tool for simulating biological neural networks at a detailed compartment level. The underlying models of dendrites axons synapses and network connectivity are presented. Accuracy of numerical results is discussed.

Nodus a software package for simulation and development of compartmental models of neurons and small networks is described. Nodus is implemented on Apple Macintosh microcomputers with the standard user interface and graphics. Two user interface issues are discussed: control of model consistency and selection of model parameters.

We describe a software package that allows for efficient simulation of current flow through complex neurons. Each neuron is represented as a binary branched tree structure,where branches are constructed from linear strings of compartments. The program is set up to allow the user to simulate typical electrophysiological experimental protocols.The modeling is done at the level of currents and voltages in individual compartments of neurons. An implicit scheme of integration which takes advantage of the branched tree structure of the neuron is used to update the voltages and currents in each neuron at each timestep. One of the major computational benefits of this method is that time scales linearly with the number of compartments used to represent the neurons. Furthermore, voltage updates are decoupled from the conductance updates, so arbitrary conductances or synaptic connections can be incorporated easily, efficiently and stably. Nemosys can also be extended to allow simulations of networks of neurons.

Programs designed specifically to simulate nerve equations compare favorably with general purpose simulation programs in three areas. 1) The user deals directly with concepts that are familiar at the neuroscience level and is not required to translate the problem into another domain. 2) The program contains functions better suited for controlling the simulation and graphing the results of real neurophysiological problems. 3) Special methods and tricks can be used to take advantage of the structure of nerve equations to solve them much more quickly, e.g. Hines (1984) and Mascagni (1991).

Activity-dependent competitive mechanisms of synaptic plasticity appear to play an important role in many processes of late neural development where an initially rough connectivity pattern refines to a precise mature pattern. A prominent example is the formation of ocular dominance columns in the visual cortex of many mammals. These processes may be modeled at several levels. Simple models use abstract neurons and assume synaptic modification according to a hebbian or similar correlation-based rule. These models incorporate biological constraints and attempt to predict large-scale developmental patterns from the combination of synaptic-level plasticity rules and measurable biological patterns of activation and connectivity. More detailed models attempt to incorporate various levels of biophysical realism including membrane and channel properties and dendritic geometry. Abstract models examine the connectivity patterns that may result if biological development follows certain dynamical or other abstract rules without concern for how such rules might be implemented at the synapse. The strengths and weaknesses of these approaches are examined through study of models for the development of ocular dominance and of orientation selectivity in the visual cortex.

The cerebral cortex of mammals can be subdivided into many different areas based on differences in internal structure, functional properties, and behavioral role. We are interested in the way these differences come about developmentally, and in how they are represented in terms of the actual cortical neural networks. One intriguing possibility is that the identity of a cortical area is determined by the type of input that it receives during its early development. To address this question, we have utilized a preparation in which we can change the modality of the input that a cortical area receives, before the stage in the animal’s life when that area has established its final identity. Specifically, we have induced visual inputs to innervate the auditory pathway, and subsequently examined the anatomical, physiological, and behavioral consequences of this rewiring of the auditory cortex.

The detection of movement by visual systems begins with the time varying signals of the photoreceptor array and critically depends upon a neuronal integrative step that compares the output of at least two sampling points of the receptor array. The receptors are approximately linear for continuously modulated illumination. Their response to a moving point of light may be calculated from the convolution of a dynamic transfer function( with constant τ)and the spatial receptive field profile (a Gaussian with acceptance angle Ø). The ratio Ø/τdetermines the relative velocity sensitivity. The tangential cells of the distal medulla externa (second optic neuropile) are the most peripheral elements in the visual pathway exhibiting directional selectivity (DS). Certain receptive field properties and antagonistic synaptic inputs from cholinergic and GABAergic pathways provide a basis for simulating DS. Our simulation is based upon an asymmetric lateral inhibition model that requires a delays in the inhibitory pathway and a separation ep between sampling stations. For a single bar with a Gaussian intensity profile DS is maximal when the velocity is Δp/ τ.For sinusoidal gratings a correct DS requires that the spatial wavelength exceed 2x Δp and that the contrast frequency CF<0.5/τ.

For a given light intensity distribution I = f(x,y)representing a scene, the visual system seems to encode the point information in terms of intensity I and in terms of the second derivative of intensity (Mach 1868). The question of where in the visual systems and in what form this encoding first takes place and what are the responsible synaptic pathways and mechanisms remains unclear. In this paper we focus on physiological mechanism underlying receptive field of a cone in the vertebrate retina and we test the data with analytical model using UCLA SFINX [1] simulation environment.

The cerebral cortex of fresh water turtles(Pseudemys scriptaandChrysemys picta)contains a visual area (review: Ulinski, 1990) that receives direct input from the dorsal lateral geniculate complex (Hall and Ebner, 1970; Heller and Ulinski, 1987). The projection of the retina to the geniculate complex is bilateral and topographically organized (Ulinski and Nautiyal, 1987), but the subsequent projection of the geniculate to the visual cortex is not topographically organized. Instead, there is a convergence of all points of visual space onto each cortical cell as a result of the spatial organization of the geniculocortical and intracortical projections (Mulligan and Ulinski, 1990; Cosans and Ulinski, 1990). The anatomy of the geniculocortical pathway is consistent with extracellular recordings in alert, paralyzed animals presented with visual stimuli (Mazurskaya, 1974). Each cortical cell in such experiments responds to moving stimuli at all points in binocular visual space. They also respond well to two stimuli presented in succession at disjunct points in visual space in an “apparent motion” paradigm and have temporal tuning curves with two facilitatory peaks, one quite constant peak at 50–75 msec and a second, more variable peak occuring between 250 and 600 msec. These neuronal properties suggest that visual cortex plays a role in visually guided orienting behaviors in turtles (Ulinski et al., 1991a).

The echolocating bat, Eptesicus fuscus emits frequency-modulated (FM) sonar signals and perceives target range from echo delay. Reconstruction of the bat’s image of a point-target from behavioral data yields approximately the ambiguity (crosscorrelation) function of echoes with additional sharpening of the function’ s central peak according to the signal-to-noise ratio of echoes. Bats perceive a shift in the phase of this function even though phase conventionally is thought to be discarded during auditory encoding of sounds in the ultrasonic frequency band used by the bat (20 to 100 kHz). The perceived range of target images undergoes amplitude-latency trading indicating that the timing of neural discharges encodes the delay of echoes including echo phase information. The bat’s auditory code resembles a spectrogram of the transmitted and the received signals in which the instantaneous frequency of FM sweeps is represented by the timing of neural discharges to each frequency in succession. A computational model of echo processing using physiologically identified peripheral auditory mechanisms indeed can produce the crosscorrelation function of echoes from a spectrogram representation. The model correlates transmission and echo spectrograms to determine target range and as a parallel process it also can transform the echo spectrum back into the time domain to estimate target shape. The combined algorithms carry out the operations of Spectrogram Correlation and Transformation (SCAT). The critical feature of auditory coding for retention of phase information in the spectrograms is half-wave-rectification of excitation at the receptor-nerve interface.

Many primary auditory neurons do not follow the temporal fine structure of waveforms within their passband; these are termed non-phase locked cells. Such cells do not encode the sound pressure waveform directly but rather some nonlinear functional of the waveform such as the envelope. But which definition of the envelope — or more generally which nonlinear functional — does a particular cell implement? This difficulty is one example of a general problem: How do we characterize information transfer in a cell which codes the output of some unknown nontrivial computation on the sensory stimulus? Here we suggest an approach in which one chooses from a set of possible nonlinear functionals by finding that functional of the stimulus which can be estimated most accurately from observation of the spike train. We apply these ideas to non-phase locked auditory afferents to measure the tuning of these cells for complex stimuli.

A model is presented of the neural computation of binaural intensity difference (BID) in the lateral shell of the central nucleus of the inferior colliculus (ICL) of the barn owl. The ICL is the second stage of binaural intensity processing following the nucleus ventralis lemnisci lateralis pars posterior (VLVp) and it projects to the external nucleus of the inferior colliculus (ICX). The model shows how monaural excitation from the cochlear nucleus angularis can be combined with inhibition from the VLVp to form a peak-like pattern of activation in the ICL from the wedge-like activation pattern in the VLVp. The model explains the functional roles of the two narrowly frequency-tuned cell types in the ICL and it predicts their patterns of interconnection. An experiment was performed in which changes in BID selectivity in the ICX were measured following the injection of neural activity modulators into the VLVp. Model simulations of these experiments produce qualitatively similiar results.

The auditory system of the barn owl (tyto alba) contains neurons sensitive to the phase of sounds of remarkably high frequency up to 9 kHz. Nucleus Laminaris represents phase differences as part of the computation of stimulus azimuth. The high frequency of the stimulus and the high level of noise in the input spike trains make the response properties of laminaris neurons hard to explain. We use simulations and semi-numerical analysis to show that the cellular and synaptic time constants must be unreasonably fast in order for ordinary biophysical mechanisms to reproduce the observed behavior. Several people have suggested that a resonance mechanism may exist in laminaris neurons to amplify the signal. We present a simple neuronal resonance model that can deal with realistic input.

The medial geniculate body of the thalamus receives input from a variety of sources, both auditory and non-auditory. The ventral division (MGv), however, appears to be a purely auditory center, that is, it receives from lower (and higher) centers which are primarily involved in encoding auditory stimuli. It has some of the characteristics of auditory cortex, e.g., layering of neurons and EE/EI bands, but has a much simpler architecture. The MGv plays a central role in analysing auditory stimuli. A wealth of data is available now on its anatomy, connectivity and responses. A detailed computational model of the ventral division of the cat medial geniculate body has been constructed. This model is being compared to physiological recordings for validation purposes, and should prove a useful tool for testing theories of auditory sensory representation and functioning.

Psychophysical acuity for speech sounds with differing voice onset time (VOT) has often been found to be nonmonotonic (Abramson and Lisker, 1970; Keating et al., 1981; Kuhl, 1981; Rosner, 1984; MacMillan et al., 1988; but see Kewley-Port et al., 1988). Acuity for VOT is usually best for the syllables that bracket the boundary between voiced (/b, d, or g/) and voiceless consonants (/p, t, or k/) (Abramson and Lisker, 1970; Kuhl, 1981). This result was once thought to reflect the operation of a neural system that was specialized for processing speech. However, it is now known that chinchillas exhibit the same relation between identification and acuity for VOT (Kuhl and Miller, 1978; Kuhl, 1981). It has been suggested that some continua of speech sounds, such as those differing in VOT, may interact with general auditory mechanisms in a way that naturally produces a region of high sensitivity (Miller, 1977; Stevens, 1989). Until recently, however, specific hypotheses about the nature of this interaction or about neural responses that underlie enhanced psychophysical acuity or category formation had not been proposed.

Studies of the neuronal organization of the jamming avoidance response (JAR) have revealed a distributed system of local computations of sensory information. A variety of electrosensory behaviors involved in social communication and object detection share networks with the JAR at the sensory and at the motor level within the hindbrain and the midbrain. Their respective flows of information however separate in the diencephalons where nodes dedicated to a single type of behavior provide sensory-motor interfaces. A comparison of neurons across different levels of sensory information processing reveals a gradual increase in the specificity and sensitivity of their responses to stimulus patterns guiding the JAR. Neurons with response properties similar in specificity and sensitivity to those of the intact behavior are encountered ultimately in the diencephalon.

The ‘small cell’ is a descriptive designation given an identified cell type in lamina 6 of the torus semicircularis of the weakly electric fish Eigenmannia that behaves as a coincidence detector similar to the cells of nucleus laminaris of the owl. A compartmental model of the small cell with Hodgkin-Huxley channel kinetics was used to explore the origin of the cell’s temporal disparity sensitivity. This model was compared with a system level model of the time disparity detecting cell of the nucleus laminaris. The small cell has no identified inhibitory synaptic input yet will fire in response to a single input and cease firing when a second one is added at certain phase differences, a functional inhibitory effect. Simulations showed that such an effect could occur due to the internal inhibition provided by voltage-sensitive potassium channels. An initial subthreshold excitatory posts ynaptic potential (EPSP) can thereby increase the threshold for a subsequent EPSP. In contrast to the n. laminaris model, this model shows a abrupt changes in rate at certain time disparities. This characteristic is not seen in the real cell and detracts from the cell’s ability to code arbitrary disparities. Obtaining a small cell response that varies smoothly with input will require either adding noise or adding more complex dynamics to the model. These alternatives could be assessed physiologically with current-clamp.

Compartmental models of electrotonic (passive) voltage spread in neural structures depend upon the morphology of the structures being modeled and the electrical characteristics of the cell. The primary morphological parameters are the length and diameter of each compartment. The electrical characteristics include membrane resistance and capacitance, cytoplasmic resistance, and conductance and time course of membrane channels. The accuracy of these model parameters determines how well the model can reflect the actual operation of the functioning neural structure.

Previous research has suggested the possible widespread relevance of chaotic dynamics to the functioning of higher-ordered neural systems. Using a multilayer model of primary and secondary olfactory structures we have demonstrated the generation of such chaotic activity. We are presently examining the characteristics of these dynamics and exploring the implications for pattern recognition applications.

Experiments on the Lateral Gastric (LG) motor neuron of the crab stomatogastric ganglion demonstrate a serotonin-sensitive region of the axonal membrane. This serotonin-sensitive region becomes a spike initiation site and propagates action potentials antidromically into the integrating regions of the stomatogastric ganglion where they do not ordinarily evoke transmitter release. A compartmental model of the neuron which includes a serotonin-sensitive slow inward current on the axon successfully simulates the physiological data. A mathematical study of axonal membranes that include slow currents provides insight into the mechanisms by which spike initiation zones can travel as waves down axons that contain slow currents.

The presence of a variety of voltage-gated channels which are activated at subthreshold membrane potentials increases the complexity of the computations that can be expected from individual neurons. This is especially so for neurons receiving large numbers of relatively weak synaptic inputs so that many spatially distributed inputs are required to excite each cell. In a model of one such neuron the spiny projection cell of the mammalian neostriatum inclusion of a simple and very common membrane nonlinearity the time-independent inwardly rectifying potassium conductance caused spatially distributed weak synapses to summate cooperatively while having no effect on the integration of spatially clustered synaptic input.

The vestibulo-ocular reflex which stabilizes the eyes during head rotation is an elegant sensorimotor system that has long attracted the attention of theoretical neuroscientists. Previous models based on classical analysis and control systems theory provided precise descriptions of overall vestibulo-ocular reflex dynamics but indicated little about the nature of the neurons that mediate this response. More recently dynamic neural network models have extended the description to the neural level providing new insights into the mechanisms of sensorimotor signal processing in the vestibulo-ocular reflex.

We present evidence for the generation of stable convergent force field patterns and muscle synergies in the spinal cord of the frog. These synergies may form the bases of postural and trajectory adjustments. We present recent analyses which show that (1) similarly structured force fields underlie natural behaviors (2) the active fields underlying both microstimulation and natural behavioral force fields are structurally invariant but may be modulated in overall force amplitude and stiffness (3) these fields are drawn from a limited set of such fields (4) these invariant fields can be used to predict the termination position of limb endpoint trajectories in the unrestrained limb (5) in those instances tested multiple stimulations resulted in fields proportional to the vectorial summation of the fields resulting from single stimulations. This recent body of work suggests that we may view the spinal cord as possessing a small number of movement primitives: circuits that may specify invariant force fields which may be combined in a more or less flexible manner to produce adjustable behaviors.

Experimental results from several laboratories over the past twenty years have suggested that the cerebellum is necessary for the learning and memory of the classically conditioned rabbit nictitating membrane response (NMR). In particular, results from R.F. Thompson’s laboratory strongly suggest that the conditional stimulus (CS) pathway involves the mossy fibres (mf), the unconditional stimulus (US) involves the climbing fibres (cf),and associated neurons in the cerebellar cortex and cerebellar nuclei form a network for the main neuronal substrates of this basic associative learninglmemory paradigm. Thompson has proposed a schematic circuit involving these cerebellar neurons and associated brain stem neurons. Blum and Thompson have refined this schematic circuit into a simplified neural network confined to the cerebellum, since experiments by Thompson et al have demonstrated that the tone CS can be replaced by direct mf stimulation in the pontine nucleus (pn) and the air-puff US can be replaced by direct cf stimulation in the inferior olive (10). In this paper, we present a simplified cerebellar network based on the Blum-Thompson network and give two mathematical models of this simplified network. We present results of computer simulations of these two models. We compare the simulation results with experimental test data from Thompson’s laboratory, andfind reasonable agreement.