Quantifying the relative contributions of divisive and subtractive feedback to rhythm generation

Biological systems are characterized by a high number of interacting components. Determining the role of each component is difficult, addressed here in the context of biological oscillations. Rhythmic behavior can result from the interplay of positive feedback that promotes bistability between high and low activity, and slow negative feedback that switches the system between the high and low activity states. Many biological oscillators include two types of negative feedback processes: divisive (decreases the gain of the positive feedback loop) and subtractive (increases the input threshold) that both contribute to slowly move the system between the high- and low-activity states. Can we determine the relative contribution of each type of negative feedback process to the rhythmic activity? Does one dominate? Do they control the active and silent phase equally? To answer these questions we use a neural network model with excitatory coupling, regulated by synaptic depression (divisive) and cellular adaptation (subtractive feedback). We first attempt to apply standard experimental methodologies: either passive observation to correlate the variations of a variable of interest to system behavior, or deletion of a component to establish whether a component is critical for the system. We find that these two strategies can lead to contradictory conclusions, and at best their interpretive power is limited. We instead develop a computational measure of the contribution of a process, by evaluating the sensitivity of the active (high activity) and silent (low activity) phase durations to the time constant of the process. The measure shows that both processes control the active phase, in proportion to their speed and relative weight. However, only the subtractive process plays a major role in setting the duration of the silent phase. This computational method can be used to analyze the role of negative feedback processes in a wide range of biological rhythms.     

Noise-induced transitions in slow wave neuronal dynamics

Many neuronal systems exhibit slow random alternations and sudden switches in activity states. Models with noisy relaxation dynamics (oscillatory, excitable or bistable) account for these temporal, slow wave, patterns and the fluctuations within states. The noise-induced transitions in a relaxation dynamics are analogous to escape by a particle in a slowly changing double-well potential. In this formalism, we obtain semi-analytically the first and second order statistical properties: the distributions of the slow process at the transitions and the temporal correlations of successive switching events. We find that the temporal correlations can be used to help distinguish among biophysical mechanisms for the slow negative feedback, such as divisive or subtractive. We develop our results in the context of models for cellular pacemaker neurons; they also apply to mean-field models for spontaneously active networks with slow wave dynamics.     

Determining the contributions of divisive and subtractive feedback in the Hodgkin-Huxley model

The Hodgkin-Huxley (HH) model is the basis for numerous neural models. There are two negative feedback processes in the HH model that regulate rhythmic spiking. The first is an outward current with an activation variable n that has an opposite influence to the excitatory inward current and therefore provides subtractive negative feedback. The other is the inactivation of an inward current with an inactivation variable h that reduces the amount of positive feedback and therefore provides divisive feedback. Rhythmic spiking can be obtained with either negative feedback process, so we ask what is gained by having two feedback processes. We also ask how the different negative feedback processes contribute to spiking. We show that having two negative feedback processes makes the HH model more robust to changes in applied currents and conductance densities than models that possess only one negative feedback variable. We also show that the contributions made by the subtractive and divisive feedback variables are not static, but depend on time scales and conductance values. In particular, they contribute differently to the dynamics in Type I versus Type II neurons.     

State transitions through inhibitory interneurons in a cortical network model

Inhibitory interneurons shape the spiking characteristics and computational properties of cortical networks. Interneuron subtypes can precisely regulate cortical function but the roles of interneuron subtypes for promoting different regimes of cortical activity remains unclear. Therefore, we investigated the impact of fast spiking and non-fast spiking interneuron subtypes on cortical activity using a network model with connectivity and synaptic properties constrained by experimental data. We found that network properties were more sensitive to modulation of the fast spiking population, with reductions of fast spiking excitability generating strong spike correlations and network oscillations. Paradoxically, reduced fast spiking excitability produced a reduction of global excitation-inhibition balance and features of an inhibition stabilised network, in which firing rates were driven by the activity of excitatory neurons within the network. Further analysis revealed that the synaptic interactions and biophysical features associated with fast spiking interneurons, in particular their rapid intrinsic response properties and short synaptic latency, enabled this state transition by enhancing gain within the excitatory population. Therefore, fast spiking interneurons may be uniquely positioned to control the strength of recurrent excitatory connectivity and the transition to an inhibition stabilised regime. Overall, our results suggest that interneuron subtypes can exert selective control over excitatory gain allowing for differential modulation of global network state.     

Theta-gamma coupling emerges from spatially heterogeneous cholinergic neuromodulation

Theta and gamma rhythms and their cross-frequency coupling play critical roles in perception, attention, learning, and memory. Available data suggest that forebrain acetylcholine (ACh) signaling promotes theta-gamma coupling, although the mechanism has not been identified. Recent evidence suggests that cholinergic signaling is both temporally and spatially constrained, in contrast to the traditional notion of slow, spatially homogeneous, and diffuse neuromodulation. Here, we find that spatially constrained cholinergic stimulation can generate theta-modulated gamma rhythms. Using biophysically-based excitatory-inhibitory (E-I) neural network models, we simulate the effects of ACh on neural excitability by varying the conductance of a muscarinic receptor-regulated K⁺ current. In E-I networks with local excitatory connectivity and global inhibitory connectivity, we demonstrate that theta-gamma-coupled firing patterns emerge in ACh modulated network regions. Stable gamma-modulated firing arises within regions with high ACh signaling, while theta or mixed theta-gamma activity occurs at the peripheries of these regions. High gamma activity also alternates between different high-ACh regions, at theta frequency. Our results are the first to indicate a causal role for spatially heterogenous ACh signaling in the emergence of localized theta-gamma rhythmicity. Our findings also provide novel insights into mechanisms by which ACh signaling supports the brain region-specific attentional processing of sensory information.     

A hemicord locomotor network of excitatory interneurons: a simulation study

Locomotor burst generation is simulated using a full-scale network model of the unilateral excitatory interneuronal population. Earlier small-scale models predicted that a population of excitatory neurons would be sufficient to produce burst activity, and this has recently been experimentally confirmed. Here we simulate the hemicord activity induced under various experimental conditions, including pharmacological activation by NMDA and AMPA as well as electrical stimulation. The model network comprises a realistic number of cells and synaptic connectivity patterns. Using similar distributions of cellular and synaptic parameters, as have been estimated experimentally, a large variation in dynamic characteristics like firing rates, burst, and cycle durations were seen in single cells. On the network level an overall rhythm was generated because the synaptic interactions cause partial synchronization within the population. This network rhythm not only emerged despite the distributed cellular parameters but relied on this variability, in particular, in reproducing variations of the activity during the cycle and showing recruitment in interneuronal populations. A slow rhythm (0.4–2 Hz) can be induced by tonic activation of NMDA-sensitive channels, which are voltage dependent and generate depolarizing plateaus. The rhythm emerges through a synchronization of bursts of the individual neurons. A fast rhythm (4–12 Hz), induced by AMPA, relies on spike synchronization within the population, and each burst is composed of single spikes produced by different neurons. The dynamic range of the fast rhythm is limited by the ability of the network to synchronize oscillations and depends on the strength of synaptic connections and the duration of the slow after hyperpolarization. The model network also produces prolonged bouts of rhythmic activity in response to brief electrical activations, as seen experimentally. The mutual excitation can sustain long-lasting activity for a realistic set of synaptic parameters. The bout duration depends on the strength of excitatory synaptic connections, the level of persistent depolarization, and the influx of Ca²⁺ ions and activation of Ca²⁺-dependent K⁺ current.     

Emergent Oscillations in a Realistic Network: The Role of Inhibition and the Effect of the Spatiotemporal Distribution of the Input

We have simulated a network of 10,000 two-compartment cells, spatially distributed on a two-dimensional sheet; 15% of the cells were inhibitory. The input to the network was spatially delimited. Global oscillations frequently were achieved with a simple set of connectivity rules. The inhibitory neurons paced the network, whereas the excitatory neurons amplified the input, permitting oscillations at low-input intensities. Inhibitory neurons were active over a greater area than excitatory ones, forming a ring of inhibition. The oscillation frequency was modulated to some extent by the input intensity, as has been shown experimentally in the striate cortex, but predominantly by the properties of the inhibitory neurons and their connections: the membrane and synaptic time constants and the distribution of delays.In networks that showed oscillations and in those that did not, widely distributed inputs could lead to the specific recruitment of the inhibitory neurons and to near zero activity of the excitatory cells. Hence the spatial distribution of excitatory inputs could provide a means of selectively exciting or inhibiting a target network. Finally, neither the presence of oscillations nor the global spike activity provided any reliable indication of the level of excitatory output from the network.     

Multi-Stability and Pattern-Selection in Oscillatory Networks with Fast Inhibition and Electrical Synapses

A model or hybrid network consisting of oscillatory cells interconnected by inhibitory and electrical synapses may express different stable activity patterns without any change of network topology or parameters, and switching between the patterns can be induced by specific transient signals. However, little is known of properties of such signals. In the present study, we employ numerical simulations of neural networks of different size composed of relaxation oscillators, to investigate switching between in-phase (IP) and anti-phase (AP) activity patterns. We show that the time windows of susceptibility to switching between the patterns are similar in 2-, 4- and 6-cell fully-connected networks. Moreover, in a network (N = 4, 6) expressing a given AP pattern, a stimulus with a given profile consisting of depolarizing and hyperpolarizing signals sent to different subpopulations of cells can evoke switching to another AP pattern. Interestingly, the resulting pattern encodes the profile of the switching stimulus. These results can be extended to different network architectures. Indeed, relaxation oscillators are not only models of cellular pacemakers, bursting or spiking, but are also analogous to firing-rate models of neural activity. We show that rules of switching similar to those found for relaxation oscillators apply to oscillating circuits of excitatory cells interconnected by electrical synapses and cross-inhibition. Our results suggest that incoming information, arriving in a proper time window, may be stored in an oscillatory network in the form of a specific spatio-temporal activity pattern which is expressed until new pertinent information arrives.     

Emergence of Slow-Switching Assemblies in Structured Neuronal Networks

Unraveling the interplay between connectivity and spatio-temporal dynamics in neuronal networks is a key step to advance our understanding of neuronal information processing. Here we investigate how particular features of network connectivity underpin the propensity of neural networks to generate slow-switching assembly (SSA) dynamics, i.e., sustained epochs of increased firing within assemblies of neurons which transition slowly between different assemblies throughout the network. We show that the emergence of SSA activity is linked to spectral properties of the asymmetric synaptic weight matrix. In particular, the leading eigenvalues that dictate the slow dynamics exhibit a gap with respect to the bulk of the spectrum, and the associated Schur vectors exhibit a measure of block-localization on groups of neurons, thus resulting in coherent dynamical activity on those groups. Through simple rate models, we gain analytical understanding of the origin and importance of the spectral gap, and use these insights to develop new network topologies with alternative connectivity paradigms which also display SSA activity. Specifically, SSA dynamics involving excitatory and inhibitory neurons can be achieved by modifying the connectivity patterns between both types of neurons. We also show that SSA activity can occur at multiple timescales reflecting a hierarchy in the connectivity, and demonstrate the emergence of SSA in small-world like networks. Our work provides a step towards understanding how network structure (uncovered through advancements in neuroanatomy and connectomics) can impact on spatio-temporal neural activity and constrain the resulting dynamics.     

Emergent oscillations in a realistic network: the role of inhibition and the effect of the spatiotemporal distribution of the input.

We have simulated a network of 10,000 two-compartment cells, spatially distributed on a two-dimensional sheet; 15% of the cells were inhibitory. The input to the network was spatially delimited. Global oscillations frequently were achieved with a simple set of connectivity rules. The inhibitory neurons paced the network, whereas the excitatory neurons amplified the input, permitting oscillations at low-input intensities. Inhibitory neurons were active over a greater area than excitatory ones, forming a ring of inhibition. The oscillation frequency was modulated to some extent by the input intensity, as has been shown experimentally in the striate cortex, but predominantly by the properties of the inhibitory neurons and their connections: the membrane and synaptic time constants and the distribution of delays. In networks that showed oscillations and in those that did not, widely distributed inputs could lead to the specific recruitment of the inhibitory neurons and to near zero activity of the excitatory cells. Hence the spatial distribution of excitatory inputs could provide a means of selectively exciting or inhibiting a target network. Finally, neither the presence of oscillations nor the global spike activity provided any reliable indication of the level of excitatory output from the network.     

Differential regulation of neuronal excitability by nicotine and substance P in subdivisions of the medial habenula

The medial habenula (MHb) plays an important role in nicotine-related behaviors, such as aversion and withdrawal. The MHb is composed of distinct subregions with unique neurotransmitter expression and neuronal connectivity. Here, we showed that nicotine and substance P (SP) differentially regulate neuronal excitability in subdivisions of the MHb (ventrolateral division, MHbVL; dorsal division; MHbD and superior division: MHbS). Nicotine remarkably increased spontaneous neuronal firing in the MHbVL and MHbD, but not in the MHbS, which was consistent with different magnitudes of whole-cell inward currents evoked by nicotine in each subdivision. Meanwhile, SP enhanced neuronal excitability in the MHbVL and MHbS. Although the MHbD is composed of SP-expressing neurons, they did not respond to SP. Neurons in the MHbVL increased their firing in response to bath-applied nicotine, which was attenuated by neurokinin receptor antagonists. Furthermore, nicotine addiction and withdrawal attenuated and augmented excitatory SP effects in the MHbVL, respectively. On the whole, we suggest that MHb-involving nicotine-related behaviors might be associated with SP signaling in MHb subdivisions.     

Effect of stretching on the action potentials of the human and rabbit ventricular myocardium

The effect of stretching from L0 to Lmax on the electrical activity was studied on human myocardial preparations from patients with heart disease and on strips of rabbit ventricular myocardium. Muscular deformation was shown to decrease the amplitude and velocity of depolarization in slow action potentials. The action potentials (AP) possessing a fast depolarization phase were not sensitive to physiological stretching. Antiarrhythmic drugs--ethmozin (2 X 10(-5) M) and ethacizin (2 X 10(-6) M)--caused a decrease in the rate of AP depolarization, thus increasing AP sensitivity to deformation. It is suggested that stretching under the action of ethmozin and ethacizin reduced cardiomyocyte excitability due to suppression of slow Ca-current.     

Gain modulation from background synaptic input

Gain modulation is a prominent feature of neuronal activity recorded in behaving animals, but the mechanism by which it occurs is unknown. By introducing a barrage of excitatory and inhibitory synaptic conductances that mimics conditions encountered in vivo into pyramidal neurons in slices of rat somatosensory cortex, we show that the gain of a neuronal response to excitatory drive can be modulated by varying the level of "background" synaptic input. Simultaneously increasing both excitatory and inhibitory background firing rates in a balanced manner results in a divisive gain modulation of the neuronal response without appreciable signal-independent increases in firing rate or spike-train variability. These results suggest that, within active cortical circuits, the overall level of synaptic input to a neuron acts as a gain control signal that modulates responsiveness to excitatory drive.     

Inter-Network Interactions: Impact of Connections between Oscillatory Neuronal Networks on Oscillation Frequency and Pattern

Oscillations in electrical activity are a characteristic feature of many brain networks and display a wide variety of temporal patterns. A network may express a single oscillation frequency, alternate between two or more distinct frequencies, or continually express multiple frequencies. In addition, oscillation amplitude may fluctuate over time. The origin of this complex repertoire of activity remains unclear. Different cortical layers often produce distinct oscillation frequencies. To investigate whether interactions between different networks could contribute to the variety of oscillation patterns, we created two model networks, one generating on its own a relatively slow frequency (20 Hz; slow network) and one generating a fast frequency (32 Hz; fast network). Taking either the slow or the fast network as source network projecting connections to the other, or target, network, we systematically investigated how type and strength of inter-network connections affected target network activity. For high inter-network connection strengths, we found that the slow network was more effective at completely imposing its rhythm on the fast network than the other way around. The strongest entrainment occurred when excitatory cells of the slow network projected to excitatory or inhibitory cells of the fast network. The fast network most strongly imposed its rhythm on the slow network when its excitatory cells projected to excitatory cells of the slow network. Interestingly, for lower inter-network connection strengths, multiple frequencies coexisted in the target network. Just as observed in rat prefrontal cortex, the target network could express multiple frequencies at the same time, alternate between two distinct oscillation frequencies, or express a single frequency with alternating episodes of high and low amplitude. Together, our results suggest that input from other oscillating networks may markedly alter a network''s frequency spectrum and may partly be responsible for the rich repertoire of temporal oscillation patterns observed in the brain.     

Active dendrites,potassium channels and synaptic plasticity

The dendrites of CA1 pyramidal neurons in the hippocampus express numerous types of voltage-gated ion channel, but the distributions or densities of many of these channels are very non-uniform. Sodium channels in the dendrites are responsible for action potential (AP) propagation from the axon into the dendrites (back-propagation); calcium channels are responsible for local changes in dendritic calcium concentrations following back-propagating APs and synaptic potentials; and potassium channels help regulate overall dendritic excitability. Several lines of evidence are presented here to suggest that back-propagating APs, when coincident with excitatory synaptic input, can lead to the induction of either long-term depression (LTD) or long-term potentiation (LTP). The induction of LTD or LTP is correlated with the magnitude of the rise in intracellular calcium. When brief bursts of synaptic potentials are paired with postsynaptic APs in a theta-burst pairing paradigm, the induction of LTP is dependent on the invasion of the AP into the dendritic tree. The amplitude of the AP in the dendrites is dependent, in part, on the activity of a transient, A-type potassium channel that is expressed at high density in the dendrites and correlates with the induction of the LTP. Furthermore, during the expression phase of the LTP, there are local changes in dendritic excitability that may result from modulation of the functioning of this transient potassium channel. The results support the view that the active properties of dendrites play important roles in synaptic integration and synaptic plasticity of these neurons.     

Coexistence of lateral and co-tuned inhibitory configurations in cortical networks

The responses of neurons in sensory cortex depend on the summation of excitatory and inhibitory synaptic inputs. How the excitatory and inhibitory inputs scale with stimulus depends on the network architecture, which ranges from the lateral inhibitory configuration where excitatory inputs are more narrowly tuned than inhibitory inputs, to the co-tuned configuration where both are tuned equally. The underlying circuitry that gives rise to lateral inhibition and co-tuning is yet unclear. Using large-scale network simulations with experimentally determined connectivity patterns and simulations with rate models, we show that the spatial extent of the input determined the configuration: there was a smooth transition from lateral inhibition with narrow input to co-tuning with broad input. The transition from lateral inhibition to co-tuning was accompanied by shifts in overall gain (reduced), output firing pattern (from tonic to phasic) and rate-level functions (from non-monotonic to monotonically increasing). The results suggest that a single cortical network architecture could account for the extended range of experimentally observed response types between the extremes of lateral inhibitory versus co-tuned configurations.     

Incorporating anatomically realistic cellular-level connectivity in neural network models of the rat hippocampus

The specific connectivity patterns among neuronal classes can play an important role in the regulation of firing dynamics in many brain regions. Yet most neural network models are built based on vastly simplified connectivity schemes that do not accurately reflect the biological complexity. Taking the rat hippocampus as an example, we show here that enough quantitative information is available in the neuroanatomical literature to construct neural networks derived from accurate models of cellular connectivity. Computational simulations based on this approach lend themselves to a direct investigation of the potential relationship between cellular connectivity and network activity. We define a set of fundamental parameters to characterize cellular connectivity, and are collecting the related values for the rat hippocampus from published reports. Preliminary simulations based on these data uncovered a novel putative role for feedforward inhibitory neurons. In particular, "mopp" cells in the dentate gyrus are suitable to help maintain the firing rate of granule cells within physiological levels in response to a plausibly noisy input from the entorhinal cortex. The stabilizing effect of feedforward inhibition is further shown to depend on the particular ratio between the relative threshold values of the principal cells and the interneurons. We are freely distributing the connectivity data on which this study is based through a publicly accessible web archive (http://www.krasnow.gmu.edu/L-Neuron).