Distributed delays stabilize neural feedback systems

Phase response approaches to neural activity models with distributed delay

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.

Anatomy and Physiology of Neurons in Layer 9 of the Chicken Optic Tectum

Visual information in birds is to great extent processed in the optic tectum (TeO), a prominent laminated midbrain structure. Retinal input enters the TeO in its superficial layers, while output is limited to intermediate and deeper layers. In addition to visual information, the TeO receives multimodal input from the auditory and somatosensory pathway. The TeO gives rise to a major ascending tectofugal projection where neurons of tectal layer 13 project to the thalamic nucleus rotundus, which then projects to the entopallium. A second tectofugal projection system, called the accessory pathway, has however not been studied as thoroughly. Again, cells of tectal layer 13 form an ascending projection that targets a nucleus known as either the caudal part of the nucleus dorsolateralis posterior of the thalamus (DLPc) or nucleus uveaformis (Uva). This nucleus is known for multimodal integration and receives additional input from the lateral pontine nucleus (PL), which in turn receives projections from layer 8–15 of the TeO. Here, we studied a particular cell type afferent to the PL that consists of radially oriented neurons in layer 9. We characterized these neurons with respect to their anatomy, their retinal input, and the modulation of retinal input by local circuits. We found that comparable to other radial neurons in the tectum, cells of layer 9 have columnar dendritic fields and reach up to layer 2. Sholl analysis demonstrated that dendritic arborization concentrates on retinorecipient layers 2 and 4, with additional arborization in layers 9 and 10. All neurons recorded in layer 9 received retinal input via glutamatergic synapses. We analyzed the influence of modulatory circuits of the TeO by application of antagonists to γ-aminobutyric acid (GABA) and acetylcholine (ACh). Our data show that the neurons of layer 9 are integrated in a network under strong GABAergic inhibition, which is controlled by local cholinergic activation. Output to the PL and to the accessory tectofugal pathway thus appears to be under strict control of local tectal networks, the relevance of which for multimodal integration is discussed.

Delay models for the early embryonic cell cycle oscillator

Time delays are known to play a crucial role in generating biological oscillations. The early embryonic cell cycle in the frog Xenopus laevis is one such example. Although various mathematical models of this oscillating system exist, it is not clear how to best model the required time delay. Here, we study a simple cell cycle model that produces oscillations due to the presence of an ultrasensitive, time-delayed negative feedback loop. We implement the time delay in three qualitatively different ways, using a fixed time delay, a distribution of time delays, and a delay that is state-dependent. We analyze the dynamics in all cases, and we use experimental observations to interpret our results and put constraints on unknown parameters. In doing so, we find that different implementations of the time delay can have a large impact on the resulting oscillations.

Synchronization-desynchronization transitions in complex networks: an interplay of distributed time delay and inhibitory nodes

We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart-Landau oscillators. To this end a symmetric network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio, synchronization tends to be unstable. However, increasing distributional widths can counteract this trend, leading to multiple resynchronization transitions at relatively high inhibition ratios. The extended applicability of the results is confirmed by numerical studies on asymmetrically perturbed network topologies. All investigations are performed on two distribution types, a uniform distribution and a Γ distribution.

Laminar segregation of GABAergic neurons in the avian nucleus isthmi pars magnocellularis: a retrograde tracer and comparative study

The isthmic complex is part of a visual midbrain circuit thought to be involved in stimulus selection and spatial attention. In birds, this circuit is composed of the nuclei isthmi pars magnocellularis (Imc), pars parvocellularis (Ipc), and pars semilunaris (SLu), all of them reciprocally connected to the ipsilateral optic tectum (TeO). The Imc conveys heterotopic inhibition to the TeO, Ipc, and SLu via widespread γ-aminobutyric acid (GABA)ergic axons that allow global competitive interactions among simultaneous sensory inputs. Anatomical studies in the chick have described a cytoarchitectonically uniform Imc nucleus containing two intermingled cell types: one projecting to the Ipc and SLu and the other to the TeO. Here we report that in passerine species, the Imc is segregated into an internal division displaying larger, sparsely distributed cells, and an external division displaying smaller, more densely packed cells. In vivo and in vitro injections of neural tracers in the TeO and the Ipc of the zebra finch demonstrated that neurons from the external and internal subdivisions project to the Ipc and the TeO, respectively, indicating that each Imc subdivision contains one of the two cell types hodologically defined in the chick. In an extensive survey across avian orders, we found that, in addition to passerines, only species of Piciformes and Rallidae exhibited a segregated Imc, whereas all other groups exhibited a uniform Imc. These results offer a comparative basis to investigate the functional role played by each Imc neural type in the competitive interactions mediated by this nucleus.

Spatiotemporal analysis of electrically evoked activity in the chicken optic tectum: a VSDI study

The midbrain is an important processing area for sensory information in vertebrates. The optic tectum and its mammalian counterpart, the superior colliculus, receive multimodal, topographic information and contain a sensory map that plays a role in spatial attention and orientation movements. Many studies have investigated the tectal circuitry by cytochemistry and by characterization of particular cell types. However, only a few studies have investigated network activation throughout the depth of the tectum. Our study provides the first data on spatiotemporal activity profiles in the depth and width of the avian optic tectum. We used an optical imaging approach with voltage-sensitive dyes to investigate population responses at a high temporal and spatial resolution. With the necessary caution due to cell extension across several layers, we can thus link our findings tentatively with the general layout of the avian optic tectum. Single electrical stimuli in the retinorecipient layers 1-4 evoked a complex optical response pattern with two components: a short, strong transient response and a weaker persistent response that lasted several hundred milliseconds. The response started in layer 5 and spread within this layer before it propagated into deeper layers. This is in line with neuroanatomical and earlier physiological data. Analysis of temporal sequence and pharmacological manipulations revealed that these responses were mainly driven by postsynaptic activation. Thus tectal network responses to patterned input can be studied by voltage-sensitive dye imaging.

VLSI Implementation of a 2.8 Gevent/s Packet-Based AER Interface with Routing and Event Sorting Functionality

State-of-the-art large-scale neuromorphic systems require sophisticated spike event communication between units of the neural network. We present a high-speed communication infrastructure for a waferscale neuromorphic system, based on application-specific neuromorphic communication ICs in an field programmable gate arrays (FPGA)-maintained environment. The ICs implement configurable axonal delays, as required for certain types of dynamic processing or for emulating spike-based learning among distant cortical areas. Measurements are presented which show the efficacy of these delays in influencing behavior of neuromorphic benchmarks. The specialized, dedicated address-event-representation communication in most current systems requires separate, low-bandwidth configuration channels. In contrast, the configuration of the waferscale neuromorphic system is also handled by the digital packet-based pulse channel, which transmits configuration data at the full bandwidth otherwise used for pulse transmission. The overall so-called pulse communication subgroup (ICs and FPGA) delivers a factor 25–50 more event transmission rate than other current neuromorphic communication infrastructures.

Motion processing with wide-field neurons in the retino-tecto-rotundal pathway

The retino-tecto-rotundal pathway is the main visual pathway in non-mammalian vertebrates and has been found to be highly involved in visual processing. Despite the extensive receptive fields of tectal and rotundal wide-field neurons, pattern discrimination tasks suggest a system with high spatial resolution. In this paper, we address the problem of how global processing performed by motion-sensitive wide-field neurons can be brought into agreement with the concept of a local analysis of visual stimuli. As a solution to this problem, we propose a firing-rate model of the retino-tecto-rotundal pathway which describes how spatiotemporal information can be organized and retained by tectal and rotundal wide-field neurons while processing Fourier-based motion in absence of periodic receptive-field structures. The model incorporates anatomical and electrophysiological experimental data on tectal and rotundal neurons, and the basic response characteristics of tectal and rotundal neurons to moving stimuli are captured by the model cells. We show that local velocity estimates may be derived from rotundal-cell responses via superposition in a subsequent processing step. Experimentally testable predictions which are both specific and characteristic to the model are provided. Thus, a conclusive explanation can be given of how the retino-tecto-rotundal pathway enables the animal to detect and localize moving objects or to estimate its self-motion parameters.

Oscillatory response in a sensory network of ON and OFF cells with instantaneous and delayed recurrent connections

A neural field model with multiple cell-to-cell feedback connections is investigated. Our model incorporates populations of ON and OFF cells, receiving sensory inputs with direct and inverted polarity, respectively. Oscillatory responses to spatially localized stimuli are found to occur via Andronov-Hopf bifurcations of stationary activity. We explore the impact of multiple delayed feedback components as well as additional excitatory and/or inhibitory non-delayed recurrent signals on the instability threshold. Paradoxically, instantaneous excitatory recurrent terms are found to enhance network responsiveness by reducing the oscillatory response threshold, allowing smaller inputs to trigger oscillatory activity. Instantaneous inhibitory components do the opposite. The frequency of these response oscillations is further shaped by the polarity of the non-delayed terms.

Dynamics of driven recurrent networks of ON and OFF cells

A globally coupled network of ON and OFF cells is studied using neural field theory. ON cells increase their activity when the amplitude of an external stimulus increases, while OFF cells do the opposite given the same stimulus. Theory predicts that, without input, multiple transitions to oscillations can occur depending on feedback delay and the difference between ON and OFF resting states. Static spatial stimuli can induce or suppress global oscillations via a Andronov-Hopf bifurcation. This is the case for either polarity of such stimuli. In contrast, only excitatory inputs can induce or suppress oscillations in an equivalent network built of ON cells only even though oscillations are more prevalent in such systems. Nonmonotonic responses to local stimuli occur where responses lateral to the stimulus switch from excitatory to inhibitory as the input amplitude increases. With local time-periodic forcing, the unforced cells oscillate at twice the driving frequency via full-wave rectification mediated by the feedback. Our results agree with simulations of the neural field model, and further, qualitative agreement is found with the behavior of a network of spiking stochastic integrate-and-fire model neurons.

Generating oscillatory bursts from a network of regular spiking neurons without inhibition

Avian nucleus isthmi pars parvocellularis (Ipc) neurons are reciprocally connected with the layer 10 (L10) neurons in the optic tectum and respond with oscillatory bursts to visual stimulation. Our in vitro experiments show that both neuron types respond with regular spiking to somatic current injection and that the feedforward and feedback synaptic connections are excitatory, but of different strength and time course. To elucidate mechanisms of oscillatory bursting in this network of regularly spiking neurons, we investigated an experimentally constrained model of coupled leaky integrate-and-fire neurons with spike-rate adaptation. The model reproduces the observed Ipc oscillatory bursting in response to simulated visual stimulation. A scan through the model parameter volume reveals that Ipc oscillatory burst generation can be caused by strong and brief feedforward synaptic conductance changes. The mechanism is sensitive to the parameter values of spike-rate adaptation. In conclusion, we show that a network of regular-spiking neurons with feedforward excitation and spike-rate adaptation can generate oscillatory bursting in response to a constant input.