Neural adaptation facilitates oscillatory responses to static inputs in a recurrent network of ON and OFF cells

We investigate the role of adaptation in a neural field model, composed of ON and OFF cells, with delayed all-to-all recurrent connections. As external spatially profiled inputs drive the network, ON cells receive inputs directly, while OFF cells receive an inverted image of the original signals. Via global and delayed inhibitory connections, these signals can cause the system to enter states of sustained oscillatory activity. We perform a bifurcation analysis of our model to elucidate how neural adaptation influences the ability of the network to exhibit oscillatory activity. We show that slow adaptation encourages input-induced rhythmic states by decreasing the Andronov–Hopf bifurcation threshold. We further determine how the feedback and adaptation together shape the resonant properties of the ON and OFF cell network and how this affects the response to time-periodic input. By introducing an additional frequency in the system, adaptation alters the resonance frequency by shifting the peaks where the response is maximal. We support these results with numerical experiments of the neural field model. Although developed in the context of the circuitry of the electric sense, these results are applicable to any network of spontaneously firing cells with global inhibitory feedback to themselves, in which a fraction of these cells receive external input directly, while the remaining ones receive an inverted version of this input via feedforward di-synaptic inhibition. Thus the results are relevant beyond the many sensory systems where ON and OFF cells are usually identified, and provide the backbone for understanding dynamical network effects of lateral connections and various forms of ON/OFF responses.     

Effect of GABAergic inhibition on odorant concentration coding in mushroom body intrinsic neurons of the honeybee

Kenyon cells, the intrinsic neurons of the insect mushroom body, have the intriguing property of responding in a sparse way to odorants. Sparse neuronal codes are often invariant to changes in stimulus intensity and duration, and sparse coding often depends on global inhibition. We tested if this is the case for honeybees’ Kenyon cells, too, and used in vivo Ca²⁺ imaging to record their responses to different odorant concentrations. Kenyon cells responded not only to the onset of odorant stimuli (ON responses), but also to their termination (OFF responses). Both, ON and OFF responses increased with increasing odorant concentration. ON responses were phasic and invariant to the duration of odorant stimuli, while OFF responses increased with increasing odorant duration. Pharmacological blocking of GABA receptors in the brain revealed that ionotropic GABA_A and metabotropic GABA_B receptors attenuate Kenyon cells’ ON responses without changing their OFF responses. Ionotropic GABA_A receptors attenuated Kenyon cell ON responses more strongly than metabotropic GABA_B receptors. However, the response dynamic, temporal resolution and paired-pulse depression did not depend on GABA_A transmission. These data are discussed in the context of mechanisms leading to sparse coding in Kenyon cells.     

Modelling intrinsic electrophysiological properties of ON and OFF retinal ganglion cells

ON and OFF retinal ganglion cells (RGCs) display differences in their intrinsic electrophysiology: OFF cells maintain spontaneous activity in the absence of any input, exhibit subthreshold membrane potential oscillations, rebound excitation and burst firing; ON cells require excitatory input to drive their activity and display none of the aforementioned phenomena. The goal of this study was to identify and characterize ionic currents that explain these intrinsic electrophysiological differences between ON and OFF RGCs. A mathematical model of the electrophysiological properties of ON and OFF RGCs was constructed and validated using published patch-clamp data from isolated intact mouse retina. The model incorporates three ionic currents hypothesized to play a role in generating behaviors that are different between ON and OFF RGCs. These currents are persistent Na⁺, I _NaP, hyperpolarization-activated, I _h, and low voltage activated Ca^2 +, I _T, currents. Using computer simulations of Hodgkin-Huxley type neuron with a single compartment model we found two distinct sets of I _NaP, I _h, I _T conductances that correspond to ON and OFF RGCs populations. Simulations indicated that special properties of I _T explain the differences in intrinsic electrophysiology between ON and OFF RGCs examined here. The modelling shows that the maximum conductance of I _T is higher in OFF than in ON cells, in agreement with recent experimental data.     

Elements for a general memory structure: properties of recurrent neural networks used to form situation models

We study how individual memory items are stored assuming that situations given in the environment can be represented in the form of synaptic-like couplings in recurrent neural networks. Previous numerical investigations have shown that specific architectures based on suppression or max units can successfully learn static or dynamic stimuli (situations). Here we provide a theoretical basis concerning the learning process convergence and the network response to a novel stimulus. We show that, besides learning “simple” static situations, a nD network can learn and replicate a sequence of up to n different vectors or frames. We find limits on the learning rate and show coupling matrices developing during training in different cases including expansion of the network into the case of nonlinear interunit coupling. Furthermore, we show that a specific coupling matrix provides low-pass-filter properties to the units, thus connecting networks constructed by static summation units with continuous-time networks. We also show under which conditions such networks can be used to perform arithmetic calculations by means of pattern completion.     

Crossover inhibition in the retina: circuitry that compensates for nonlinear rectifying synaptic transmission

In the mammalian retina, complementary ON and OFF visual streams are formed at the bipolar cell dendrites, then carried to amacrine and ganglion cells via nonlinear excitatory synapses from bipolar cells. Bipolar, amacrine and ganglion cells also receive a nonlinear inhibitory input from amacrine cells. The most common form of such inhibition crosses over from the opposite visual stream: Amacrine cells carry ON inhibition to the OFF cells and carry OFF inhibition to the ON cells (”crossover inhibition”). Although these synapses are predominantly nonlinear, linear signal processing is required for computing many properties of the visual world such as average intensity across a receptive field. Linear signaling is also necessary for maintaining the distinction between brightness and contrast. It has long been known that a subset of retinal outputs provide exactly this sort of linear representation of the world; we show here that rectifying (nonlinear) synaptic currents, when combined thorough crossover inhibition can generate this linear signaling. Using simple mathematical models we show that for a large set of cases, repeated rounds of synaptic rectification without crossover inhibition can destroy information carried by those synapses. A similar circuit motif is employed in the electronics industry to compensate for transistor nonlinearities in analog circuits.     

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.     

The RF-cinematogram

We present a spike-triggered averaging method capable of mapping the visual receptive fields of several neurons simultaneously. The stimulation is general and the mapping proceeds automatically without the need to match the stimulation to the cells' preference for position, orientation, direction, etc. The maps are spatiotemporal; receptive field (RF) structures are quantitatively determined in three dimensions: the two dimensions of visuotopic space, and time. The method presented is one of a family of reverse correlation or spike-triggered averaging techniques (DeBoer and Kuyper 1968) capable of revealing linear aspects of stimulus-response coupling. The formal relationship of these methods to stimulus-response crosscorrelation is shown. The analysis is extended to provide some second-order axis-of-motion information (direction marks). The stimulus is a constantly illuminated, randomly jumping bright or dark spot, not an elongated bar. Spot diameters between one-third to 1 × RF width are effective. The method ascertains for each recorded action potential or spike the prior visual field position of the spot. The average or most probable spot positions define the receptive field spatially. Repeating the process for a succession of times prior to observed spikes defines the field temporally, presented here as a succession of spatial maps. We term this portrayal a receptive field cinematogram, RFc or ciné. The RFc reveals and economically portrays the spread of excitability and suppression across the receptive field, culminating in the generation of a spike. RFcs for LGN neurons and for simple cells recorded in cat cortical areas 17 and 18 are presented and interpreted in terms of classic ON/OFF regions. The availability of temporal information permits the separation of an excitatory exit response, generated when a moving bright spot leaves an OFF region, from an excitatory entrance response occurring when a bright spot enters an ON region, because these responses occur at different times (exit responses earlier). Spike emission remains coupled to (cross-correlated with) stimulus events over time periods as long as 96 ms, implying that some stimulus drive or afferent visual input is delayed by as much as 96 ms more than other input. This is a striking instance of temporal dispersion in the visual system. In some cells, said to be spatiotemporally inseparable, the delay (latency) varies systematically across the visual field; i.e., the place for optimal stimulation varies with the time prior to spike emission. In these cells, the RFc shows receptive field structures which move across the visual field over trajectories equal to approximately twice the total conventional RF width. Exit and entrance responses, on the other hand, arise in a simple way from separated ON and OFF RF subregions. ON/ OFF mechanisms thus appear unrelated to spatiotemporal inseparability. The RFc method is easily automated, efficient, and characterizes multiple RFs simultaneously, as required in work with multiple electrode arrays.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

Probabilistic computation by neuromine networks

In this paper, we address the question, can biologically feasible neural nets compute more than can be computed by deterministic polynomial time algorithms? Since we want to maintain a claim of plausibility and reasonableness we restrict ourselves to algorithmically easy to construct nets and we rule out infinite precision in parameters and in any analog parts of the computation. Our approach is to consider the recent advances in randomized algorithms and see if such randomized computations can be described by neural nets. We start with a pair of neurons and show that by connecting them with reciprocal inhibition and some tonic input, then the steady-state will be one neuron ON and one neuron OFF, but which neuron will be ON and which neuron will be OFF will be chosen at random (perhaps, it would be better to say that microscopic noise in the analog computation will be turned into a megascale random bit). We then show that we can build a small network that uses this random bit process to generate repeatedly random bits. This random bit generator can then be connected with a neural net representing the deterministic part of randomized algorithm. We, therefore, demonstrate that these neural nets can carry out probabilistic computation and thus be less limited than classical neural nets.     

Contribution to the theory of random and biased nets

The probabilistic theory of random and biased nets is further developed by the “tracing” method treated previously. A number of biases expected to be operating in nets, particularly in sociograms, is described. Distribution of closed chain lengths is derived for random nets and for nets with a simple “reflexive” bias. The “island model” bias is treated for the case of two islands and a single axon tracing, resulting in a pair of linear difference equations with two indices. The reflexive bias is extended to multiple-axon tracing by an approximate method resulting in a modification of the random net recursion formula. Results previously obtained are compared with empirical findings and attempts are made to account for observed discrepancies.     

Tuned solutions in dynamic neural fields as building blocks for extended EEG models

The most prominent functional property of cortical neurons in sensory areas are their tuned receptive fields which provide specific responses of the neurons to external stimuli. Tuned neural firing indeed reflects the most basic and best worked out level of cognitive representations. Tuning properties can be dynamic on a short time-scale of fractions of a second. Such dynamic effects have been modeled by localised solutions (also called “bumps” or “peaks”) in dynamic neural fields. In the present work we develop an approximation method to reduce the dynamics of localised activation peaks in systems of n coupled nonlinear d-dimensional neural fields with transmission delays to a small set of delay differential equations for the peak amplitudes and widths only. The method considerably simplifies the analysis of peaked solutions as demonstrated for a two-dimensional example model of neural feature selectivity in the brain. The reduced equations describe the effective interaction between pools of local neurons of several (n) classes that participate in shaping the dynamic receptive field responses. To lowest order they resemble neural mass models as they often form the base of EEG-models. Thereby they provide a link between functional small-scale receptive field models and more coarse-grained EEG-models. More specifically, they connect the dynamics in feature-selective cortical microcircuits to the more abstract local elements used in coarse-grained models. However, beside amplitudes the reduced equations also reflect the sharpness of tuning of the activity in a d-dimensional feature space in response to localised stimuli.     

Axonal Synapses Utilize Multiple Synaptic Ribbons in the Mammalian Retina

In the mammalian retina, bipolar cells and ganglion cells which stratify in sublamina a of the inner plexiform layer (IPL) show OFF responses to light stimuli while those that stratify in sublamina b show ON responses. This functional relationship between anatomy and physiology is a key principle of retinal organization. However, there are at least three types of retinal neurons, including intrinsically photosensitive retinal ganglion cells (ipRGCs) and dopaminergic amacrine cells, which violate this principle. These cell types have light-driven ON responses, but their dendrites mainly stratify in sublamina a of the IPL, the OFF sublayer. Recent anatomical studies suggested that certain ON cone bipolar cells make axonal or ectopic synapses as they descend through sublamina a, thus providing ON input to cells which stratify in the OFF sublayer. Using immunoelectron microscopy with 3-dimensional reconstruction, we have identified axonal synapses of ON cone bipolar cells in the rabbit retina. Ten calbindin ON cone bipolar axons made en passant ribbon synapses onto amacrine or ganglion dendrites in sublamina a of the IPL. Compared to the ribbon synapses made by bipolar terminals, these axonal ribbon synapses were characterized by a broad postsynaptic element that appeared as a monad and by the presence of multiple short synaptic ribbons. These findings confirm that certain ON cone bipolar cells can provide ON input to amacrine and ganglion cells whose dendrites stratify in the OFF sublayer via axonal synapses. The monadic synapse with multiple ribbons may be a diagnostic feature of the ON cone bipolar axonal synapse in sublamina a. The presence of multiple ribbons and a broad postsynaptic density suggest these structures may be very efficient synapses. We also identified axonal inputs to ipRGCs with the architecture described above.     

Detection and estimation of neural connectivity based on crosscorrelation analysis

Crosscorrelation analysis of simultaneously recorded activity of pairs of neurons is a common tool to infer functional neural connectivity. The adequacy of crosscorrelation procedures to detect and estimate neural connectivity has been investigated by means of computer simulations of small networks composed of fairly realistic modelneurons. If the mean interval of neural firings is much larger than the duration of postsynaptic potentials, which will be the case in many central brain areas excitatory connections are easier to detect than inhibitory ones. On the other hand, inhibitory connections are revealed better if the mean firing interval is much smaller than post-synaptic potential duration. In general the effects of external stimuli and the effects of neural connectivity do not add linearly. Furthermore, neurons may exhibit a certain degree of timelock to the stimulus. For these reasons the commonly applied shift predictor procedure to separate stimulus and neural effects appears to be of limited value. In case of parallel direct and indirect neural pathways between two neurons crosscorrelation analysis does not estimate the direct connection but instead an effective connectivity, which reflects the combined influences of the parallel pathways.Abbreviations ACH autocoincidence histogram - CCH crosscoincidence histogram - CPDF crossproduct density function - DCH difference crosscoincidence histogram - NACH nonsimultaneous autocoincidence histogram - NCCH nonsimultaneous crosscoincidence histogram - SCCH scaled crosscoincidence histogram - SDCH scaled difference crosscoincidence histogram     

Non-centered spike-triggered covariance analysis reveals neurotrophin-3 as a developmental regulator of receptive field properties of ON-OFF retinal ganglion cells

The functional separation of ON and OFF pathways, one of the fundamental features of the visual system, starts in the retina. During postnatal development, some retinal ganglion cells (RGCs) whose dendrites arborize in both ON and OFF sublaminae of the inner plexiform layer transform into RGCs with dendrites that monostratify in either the ON or OFF sublamina, acquiring final dendritic morphology in a subtype-dependent manner. Little is known about how the receptive field (RF) properties of ON, OFF, and ON-OFF RGCs mature during this time because of the lack of a reliable and efficient method to classify RGCs into these subtypes. To address this deficiency, we developed an innovative variant of Spike Triggered Covariance (STC) analysis, which we term Spike Triggered Covariance - Non-Centered (STC-NC) analysis. Using a multi-electrode array (MEA), we recorded the responses of a large population of mouse RGCs to a Gaussian white noise stimulus. As expected, the Spike-Triggered Average (STA) fails to identify responses driven by symmetric static nonlinearities such as those that underlie ON-OFF center RGC behavior. The STC-NC technique, in contrast, provides an efficient means to identify ON-OFF responses and quantify their RF center sizes accurately. Using this new tool, we find that RGCs gradually develop sensitivity to focal stimulation after eye opening, that the percentage of ON-OFF center cells decreases with age, and that RF centers of ON and ON-OFF cells become smaller. Importantly, we demonstrate for the first time that neurotrophin-3 (NT-3) regulates the development of physiological properties of ON-OFF center RGCs. Overexpression of NT-3 leads to the precocious maturation of RGC responsiveness and accelerates the developmental decrease of RF center size in ON-OFF cells. In summary, our study introduces STC-NC analysis which successfully identifies subtype RGCs and demonstrates how RF development relates to a neurotrophic driver in the retina.     

Testable predictions from realistic neural network simulations of vestibular compensation: integrating the behavioural and physiological data

Neural network simulations have been used previously in the investigation of the horizontal vestibulo-ocular reflex (HVOR) and vestibular compensation. The simulations involved in the present research were based on known anatomy and physiology of the vestibular pathway. This enabled the straightforward comparison of the network response, both in terms of behavioural (eye movement) and physiological (neural activity) data to empirical data obtained from guinea pig. The network simulations matched the empirical data closely both in terms of the static symptoms (spontaneous nystagmus) of unilateral vestibular deafferentation (UVD) as well as in terms of the dynamic symptoms (decrease in VOR gain). The use of multiple versions of the basic network, trained to simulate individual guinea pigs, highlighted the importance of the particular connections: the vestibular ganglion to the type I medial vestibular nucleus (MVN) cells on the contralesional side. It also indicated the significance of the relative firing rate in type I MVN cells which make excitatory connections with abducens cells as contributors to the variability seen in the level of compensated response following UVD. There was an absence of any difference (both in terms of behavioural and neural response) between labyrinthectomised and neurectomised simulations. The fact that a dynamic VOR gain asymmetry remained following the elimination of the spontaneous nystagmus in the network suggested that the amelioration of both the static and dynamic symptoms of UVD may be mediated by a single network. The networks were trained on high acceleration impulse stimuli but displayed the ability to generalise to low frequency, low acceleration sinusoids and closely approximated the behavioural responses to those stimuli. Received: 12 October 1998 / Accepted in revised form: 11 February 1999     

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.     

An analysis of the 2o and 10o field color-matching functions

The 2^o and 10^o field color-matching functions are independent: one specification is not a linear transformation of the other, even after correcting for macular pigment effects. Therefore, the “true” color-matching functions which directly describe the linear responses of the eye must be different for the two field sizes. This means that a given stimulus will, in general, have a different chromaticity depending upon the field size, regardless of the choice of any one colorimetric co-ordinate system for all field sizes. However, in spite of these chromaticity differences, a large uniform field usually appears nearly uniform. Such color uniformity implies that even though chromatic differences occur as a function of retinal position or field size, these differences are small. If this is the case, then the underling “true” color-matching functions determining the observed color-matching functions must be nearly, but not quite, identical. These differences vanish as identity between the sets of color-matching functions is approached. This property suggests a method of calculating the “true” color-matching functions. The “true” color-matching functions must approximate those obtained by minimizing the chromaticity differences between two independent sets of data. This can be done by assuming that the coefficients of transformation should be adjusted so as to produce as nearly identical chromaticities for spectrum stimuli as possible. In this paper, it is also assumed that the “true” color-matching functions have no negative values, as if they were based on actual absorption spectra. This article describes the calculation of the “true” 2^o and 10^o field color-matching functions satisfying these two conditions. For both field sizes, the maxima of the three functions are near 435, 540, and 585 mμ, after correcting for the filtering effects of the ocular media and macular pigment.     

The d-wave in fish and the state of light adaptation

Comparative electroretinographic studies of the d-wave evoked with long duration photo stimuli in dark- and light-adapted fish species (three marine and three freshwater) were performed. At the end of prolonged photo-stimulation in scotopic conditions a negative d-wave appears in electroretinograms of dogfish shark, eel and goldfish diminishing and eventually changing with intensity of photo-stimulation, while in rudd it only increases. Dark-adapted electroretinograms of two percids (perch and painted comber) exhibit a positive d-wave that approaches the b-wave amplitude under bright photopic conditions. Judging from the d-wave, only the rod pathway is active in dark-adapted dogfish shark, eel, and goldfish. Under the light adaptation, cone pathways are active in eel and goldfish, whereas the positive response to the end of light stimuli in dogfish shark could be explained by independent ON and OFF pathways from outer to inner retina via bipolar cells. In the case of two percids, dark adaptation has no influence on cone pathways. The d-wave of rudd behaves like cone-driven d-waves but in opposite sign. The data thus show that the d-wave form, amplitude and sign depend on interconnection of ON and OFF pathways as determined by the state of adaptation and/or type of photoreceptor.     

The perfect learner

As a “base line” of memorization performance, the behavior of a “perfect learner” is considered. He is characterized by a perfect memory and by the ability to choose the best search procedure in problems where the correct response from a given repertoire is to be found to each of several stimuli under the condition of “right” and “wroing” promptings by the experimenter. Expected learning curves are derived for the case of disjoint response repertoires associated with the stimuli under cyclic and random presentation of the stimuli and for the case of a single response repertoire (a one-to-one matching problem) under cyclic presentation.