Input output curves of aggregates of “simple counter” neurons

The refractory periods of an aggregate of simple “counter” neurons are assumed distributed according to some probability frequency. The output of the aggregate is computed for rectangular and triangular distributions. In particular, it is shown that the maximum output of an aggregate with any triangular distribution cannot exceed the maximum output of its average neuron by a factor greater than 2 ln 2. This puts an upper bound on the amount of departure from the behavior of the average neuron which an aggregate characterized by a certain type of distribution can show. Next, the aggregate is supposed to be subjected to regularly spaced stimuli. Under these conditions, a single neuron will give a discontinuous output curve. If, however, the refractory periods are distributed according to some frequency, the output curve may be “smoothed out.” A general condition on the distribution is derived which makes the output monotone increasing with the input. The condition is applied to some special cases.     

Contribution to the probabilistic theory of neural nets: II. Facilitation and threshold phenomena

The output curve of a single neuron with a threshold of response with respect to the frequency of the stimuli is derived. If the stimuli are regularly spaced in time, the output curve has discontinuities. If the threshold and/or refractory period are sufficiently large, the output curve approaches the “all-or-none” curve. In the case of completely randomized stimuli, the output curve is sigmoid. The equation of this curve is derived and some properties are studied. Threshold and “all-or-none” effects can be achieved by “pyramiding” neurons of this type to converge on neurons of higher order.     

“Addition” and “multiplication” theorems for the inputs of two neurons converging on a third

The output of a neuron innervated by two other neurons which, in turn, are subjected to two independent Poisson showers of stimuli, is derived as a function of the frequencies of the Poisson showers under two distinct assumptions, 1) where either of the two neurons can fire the third, and 2) where the stimuli from both neurons must impinge within a certain time interval to fire the third. For very small frequencies, the output of the third neuron is very nearly the sum of the input frequencies in the first case and proportional to the product of the input frequencies in the second case. Hence the designation “addition” and “multiplication” theorems. This treatment is a generalization of a previous treatment where the Poisson shower was assumed identical for the two outer neurons.     

Contribution to the probabilistic theory of neural nets: III. Specific inhibition

The input-output formula is derived for a neuron upon which converge the axones of two other neurons (one excitatory, the other inhibitory) which are themselves subjected to a “Poisson shower” of excitatory stimuli. If the period of latent inhibition, σ, does not exceed one half the refractory period, δ, the input-output curve has no maximum. If, however, σ>δ/2, a maximum exists in the input-output curve. As the outside frequencyx increases without bound, the output frequencyx ₃ approaches an asymptotic value which ranges from 1/δ to 0, depending on the ratio σ/δ. The maximum output (if it exists) is also derived as a function of σ and δ.     

Analysis of the exponential decay model of the neuron showing frequency threshold effects

The exponential decay model of a neuron has been analyzed using the “random walk” approach of stochastic processes and an “absorbing barrier” solution is obtained forg ^T (s)—the Laplace transform of the output pulse interval density function. An expression for the mean output frequency is derived from this and a variety of input-output curves plotted which show frequency threshold effects in single neurons. Our results are compared with those of other authors obtained by computer simulation techniques, and the significance of these results discussed with reference to the possible behavior of networks constructed of such neuron units.     

Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

Some implications of the short-term synaptic plasticity for neuronal computation: a model study

Spikes arriving at the synaptic connection produce short-term plastic changes of the synaptic efficacy. Model experiments have shown that paired-pulse facilitation attaining its maximum after a specific interval between a pair of arriving spikes might turn a “weak” plastic synapse attached to an integrate-and-fire neuron to a frequency-tuned device. Resulting computational capabilities create biologically plausible mechanisms of information processing relating to: (i) real-time identification of temporal patterns in a stream of random spiking activity (a recognition problem); and (ii) codetermination of the specific activity routing among neurons (an addressing problem) resulting in definite spatio-temporal patterns of the output activity (an input-output pattern problem).     

Computational simulation of the input-output relationship in hippocampal pyramidal cells

The precise mapping of how complex patterns of synaptic inputs are integrated into specific patterns of spiking output is an essential step in the characterization of the cellular basis of network dynamics and function. Relative to other principal neurons of the hippocampus, the electrophysiology of CA1 pyramidal cells has been extensively investigated. Yet, the precise input-output relationship is to date unknown even for this neuronal class. CA1 pyramidal neurons receive laminated excitatory inputs from three distinct pathways: recurrent CA1 collaterals on basal dendrites, CA3 Schaffer collaterals, mostly on oblique and proximal apical dendrites, and entorhinal perforant pathway on distal apical dendrites. We implemented detailed computer simulations of pyramidal cell electrophysiology based on three-dimensional anatomical reconstructions and compartmental models of available biophysical properties from the experimental literature. To investigate the effect of synaptic input on axosomatic firing, we stochastically distributed a realistic number of excitatory synapses in each of the three dendritic layers. We then recorded the spiking response to different stimulation patterns. For all dendritic layers, synchronous stimuli resulted in trains of spiking output and a linear relationship between input and output firing frequencies. In contrast, asynchronous stimuli evoked non-bursting spike patterns and the corresponding firing frequency input-output function was logarithmic. The regular/irregular nature of the input synaptic intervals was only reflected in the regularity of output inter-burst intervals in response to synchronous stimulation, and never affected firing frequency. Synaptic stimulations in the basal and proximal apical trees across individual neuronal morphologies yielded remarkably similar input-output relationships. Results were also robust with respect to the detailed distributions of dendritic and synaptic conductances within a plausible range constrained by experimental evidence. In contrast, the input-output relationship in response to distal apical stimuli showed dramatic differences from the other dendritic locations as well as among neurons, and was more sensible to the exact channel densities. Action Editor: Alain Destexhe     

A note on Rapoport's approximate formula for the input-output curve in the case of bilateral pre-inhibition

The accuracy of an approximation method used in a certain input-output problem of randomized stimuli is evaluated. The curves derived from it are shown to be close approximations to those derived by a statistically “exact” method.     

A note on facilitation and threshold phenomena

A neuron subjected to a Poisson shower of stimuli responds only ifh stimuli impinge upon it within the time interval ρ. It is shown that the derivative of the input-output curve cannot exceed unity.     

Conditioned reflex: detector and command neuron

Conditioned reflex is characterized by plasticity resulting in a bilateral selective input-output linking. In simple nervous systems, input stimuli are represented by selective detectors connected with command neurons through plastic synapses strengthened during associative learning and weakened during extinction. The process of associative learning is due to temporal coincidence of excitation in both detector and command neurons. Short-term memory within a plastic synapses is mediated by phosphorilation of postsynaptic receptor molecules not requiring protein synthesis. Long-term synaptic memory parallels expression of immediate early genes that mediates structural gene expression and protein synthesis. A simple detector-command neuron association becomes more complex in the course of evolution. Input mechanism is supplemented with predetector interneurons preceding detectors. Detector selectively tuned to specific input stimulus is converging on a command neuron constitute selectivity mechanism for conditioned reflexes to complex stimuli. The complication also concerns the output mechanisms. Command neurons become more specialized, and an additional link of premotor interneurons is incorporated between command neurons and motor neurons. Via synapses, the command neurons can produce excitation in a particular set of premotor neurons controlling a specific set of motor neurons responsible for behavioral act configuration. Specialization of command neurons in combination with premotor neuron structures increases the variability of outputs. Conditioned reflexes with more complex inputs and more flexible outputs determine the diversity of acquired behaviors.     

Steady states in random nets: I

A neural net is taken to consist of a semi-infinite chain of neurons with connections distributed according to a certain probability frequency of the lengths of the axones. If an input of excitation is “fed” into the net from an outside source, the statistical properties of the net determine a certain steady state output. The general functional relation between the input and the output is derived as an integral equation. For a certain type of probability distribution of connections, this equation is reducible to a differential equation. The latter can be solved by elementary methods for the output in terms of the input in general and for the input in terms of the output in special cases.     

A learned neural network that simulates properties of the neuronal population vector

This paper considers steady-state and timedependent characteristics of the response of the hidden-layer neurons in a dynamic model for the neural network trained through supervised learning to perform transformation of input signals into output signals. This transformation is set up so as to correspond to variation in the directions of two-dimensional vectors and is treated as creation by the network of a movement direction in response to a stimulus direction. The input vector is encoded in the state of the input layer at the initial instant of time, and the output vector in the state of the output layer at great values of time. After the network has been trained on examples of the input-output relation, the hidden neurons turn out to be broadly tuned to direction. The corresponding dependence for their activity is approximated with a smooth function, whose maximum allows some preferred direction to be attributed to each neuron. If each hidden neuron is assigned a vector pointing in its preferred direction, then any arbitrarily chosen direction can be characterized by an imaginary neuronal population vector (Georgopoulos et al. 1986) defined as the sum of the vectors of preferred direction for the neurons, with the weights equal to their activities for the chosen direction. It is demonstrated that, although hidden neurons are broadly tuned to direction, the population vector points in a direction congruent with that of the input vector at the initial moment of time and accurately predicts the direction of the output vector at great values of time. In between, the population vector turns continuously from the one direction towards the other. The dynamic and stationary properties of the population vector of the hidden-layer neurons, as obtained within the framework of the model in question, show a close similarity to the experimentally observed (Georgopoulos et al. 1986; Georgopoulos et al. 1989) behaviour of the population vector constructed in the same manner on the ensemble of motor cortex neurons sensitive to a certain type of movement.     

A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density

A neuron is assumed to receive synaptic input of both excitatory and inhibitory natures from a large number of neighboring neurons; it is also assumed that a large number of such impulses are required to raise the neuron’s transmembrane potential to its threshold potential, at which it “fires” or “spikes”. The model is similar to one of Gerstein and Mandelbrot, except that in the absence of input an exponential decay of potential toward a resting level is introduced. Computational methods of determining the spike timeinterval distribution are discussed, along with the inverse problem of estimating the parameters of the system from observed spike time-interval data.     

A canonical form for neural nets without circles

By “neural net” will be meant “neural net without circles.” Every neural net effects a transformation from inputs (i.e., firing patterns of the input neurons) to outputs (firing patterns of the output neurons). Two neural nets will be calledequivalent if they effect the same transformation from inputs to outputs. A canonical form is found for neural nets with respect to equivalence; i.e., a class of neural nets is defined, no two of which are equivalent, and which contains a neural net equivalent to any given neural net. This research was supported by the U.S. Air Force under Contract AF 49(638)-414 monitored by the Air Force Office of Scientific Research.     

Gamma oscillations as a mechanism for selective information transmission

In the past decades, many studies have focussed on the relation between the input and output of neurons with the aim to understand information processing by neurons. A particular aspect of neuronal information, which has not received much attention so far, concerns the problem of information transfer when a neuron or a population of neurons receives input from two or more (populations of) neurons, in particular when these (populations of) neurons carry different types of information. The aim of the present study is to investigate the responses of neurons to multiple inputs modulated in the gamma frequency range. By a combination of theoretical approaches and computer simulations, we test the hypothesis that enhanced modulation of synchronized excitatory neuronal activity in the gamma frequency range provides an advantage over a less synchronized input for various types of neurons. The results of this study show that the spike output of various types of neurons [i.e. the leaky integrate and fire neuron, the quadratic integrate and fire neuron and the Hodgkin–Huxley (HH) neuron] and that of excitatory–inhibitory coupled pairs of neurons, like the Pyramidal Interneuronal Network Gamma (PING) model, is highly phase-locked to the larger of two gamma-modulated input signals. This implies that the neuron selectively responds to the input with the larger gamma modulation if the amplitude of the gamma modulation exceeds that of the other signals by a certain amount. In that case, the output of the neuron is entrained by one of multiple inputs and that other inputs are not represented in the output. This mechanism for selective information transmission is enhanced for short membrane time constants of the neuron.     

Exclusive-OR function of single arborized neuron

We designed four arborized neurons which are able to evaluate the exclusive-or (XOR) function from two inputs. The input neurons form exclusively excitatory synapses on a dendritic tree which is a patchwork of passive (ohmic) and active cable segments. The active segments are described by the Hodgkin-Huxley model. The dynamics of the neurons and their output are obtained by numerical integration of the cable equation. In neurons 1 and 2 the XOR function is based on the annihilation of colliding action potentials. In neuron No. 3 the design takes advantage of the refractory period of action potentials. In neuron No. 4 voltage inversion is used as it occurs for inactivated sodium conductance in the Hodgkin-Huxley model. In all cases the XOR function depends critically on an appropriate timing of the input signals and on delays of the voltage transients in different branches of the dendrite.     

A locally inhibited lateral neural network: on the fundamental features of a network consisting of neurons with a restricted range of interaction

A network model that consists of neurons with a restricted range of interaction is presented. The neurons are connected mutually by inhibition weights. The inhibition of the whole network can be controlled by the range of interaction of a neuron. By this local inhibition mechanism, the present network can produce patterns with a small activity from input patterns with various large activities. Moreover, it is shown in simulation that the network has attractors for input patterns. The appearance of attractors is caused by the local interaction of neurons. Thus, we expect that the network not only works as a kind of filter, but also as a memory device for storing the produced patterns. In the present paper, the fundamental features and behavior of the network are studied by using a simple network structure and a simple rule of interaction of neurons. In particular, the relation between the interaction range of a neuron and the activity of input-output patterns is shown in simulation. Furthermore, the limit of the␣transformation and the size of basin are studied numerically. Received: 5 January 1995 / Accepted in revised form: 13 November 1997     

Some voluntary C-bends may be Mauthner neuron initiated

Predatory fish sometimes capture a prey fish first by striking it from the side, allowing the predator to consume the stunned prey head first. The rapid body flexion that the predator uses to stun its prey is similar to the “C” shaped maneuver (“C-bend”) that many fish species use when performing a C-start escape response. For most species, one of the two Mauthner neurons initiates the C-start and, together with other reticulospinal neurons, their activity determines the extent of the bend and the ultimate trajectory of the fish. Reported here is initial evidence of previously undescribed behaviors where goldfish strike an object while executing voluntary C-bends that are similar to their C-start escape responses. The overlapping distributions of turn durations, turn angles, and angular velocities suggest that at least some voluntary C-bends are initiated by the Mauthner neuron. This implies that the Mauthner neuron can be activated voluntarily in the absence of predator- or feeding-associated releasing stimuli.     

Morphological characterization of single fan-shaped body neurons in <Emphasis Type="Italic">Drosophila melanogaster</Emphasis>

The fan-shaped body is the largest substructure of the central complex in Drosophila melanogaster. Two groups of large-field neurons that innervate the fan-shaped body, viz., F1 and F5 neurons, have recently been found to be involved in visual pattern memory for “contour orientation” and “elevation” in a rut-dependent manner. The F5 neurons have been found to be responsible for the parameter “elevation” in a for-dependent manner. We have shown here that the F1 neuron also affects visual memory for “contour orientation” in a for-dependent way. With the help of Gal4/UAS and FLP-out techniques, we have characterized the morphological features of these two groups of neurons at single neuron resolution. We have observed that F1 or F5 neurons are groups of isomorphic individual neurons. Single F1 neurons have three main arborization regions: one in the first layer of the fan-shaped body, one in the ventral body, and another in the inferior medial protocerebrum. Single F5 neurons have two arborization regions: one in the fifth layer of the fan-shaped body and the other in the superior medial protocerebrum. The polarity of the F1 and F5 neurons has been studied with the Syt-GFP marker. Our results indicate the existence of presynaptic sites of both F1 and F5 neurons located in the fan-shaped body and postsynaptic sites outside of the fan-shaped body. This work was supported by the “973 Program” (2005CB522804 and 2009CB918702), the National Natural Sciences Foundation of China (30621004, 30625022, and 30770682), and the Knowledge Innovation Program of the Chinese Academy of Sciences (KSCX2-YW-R-28).