Transitions between asynchronous and synchronous states: a theory of correlations in small neural circuits

The study of correlations in neural circuits of different size, from the small size of cortical microcolumns to the large-scale organization of distributed networks studied with functional imaging, is a topic of central importance to systems neuroscience. However, a theory that explains how the parameters of mesoscopic networks composed of a few tens of neurons affect the underlying correlation structure is still missing. Here we consider a theory that can be applied to networks of arbitrary size with multiple populations of homogeneous fully-connected neurons, and we focus its analysis to a case of two populations of small size. We combine the analysis of local bifurcations of the dynamics of these networks with the analytical calculation of their cross-correlations. We study the correlation structure in different regimes, showing that a variation of the external stimuli causes the network to switch from asynchronous states, characterized by weak correlation and low variability, to synchronous states characterized by strong correlations and wide temporal fluctuations. We show that asynchronous states are generated by strong stimuli, while synchronous states occur through critical slowing down when the stimulus moves the network close to a local bifurcation. In particular, strongly positive correlations occur at the saddle-node and Andronov-Hopf bifurcations of the network, while strongly negative correlations occur when the network undergoes a spontaneous symmetry-breaking at the branching-point bifurcations. These results show how the correlation structure of firing-rate network models is strongly modulated by the external stimuli, even keeping the anatomical connections fixed. These results also suggest an effective mechanism through which biological networks may dynamically modulate the encoding and integration of sensory information.     

Temporal oscillations in neuronal nets

Summary A model for the interactions of cortical neurons is derived and analyzed. It is shown that small amplitude spatially inhomogeneous standing oscillations can bifurcate from the rest state. In a periodic domain, traveling wave trains exist. Stability of these patterns is discussed in terms of biological parameters. Homoclinic and heteroclinic orbits are demonstrated for the space-clamped system.The research reported in this paper was supported in part by NIH GM2037     

Decision making in recurrent neuronal circuits

Decision making has recently emerged as a central theme in neurophysiological studies of cognition, and experimental and computational work has led to the proposal of a cortical circuit mechanism of elemental decision computations. This mechanism depends on slow recurrent synaptic excitation balanced by fast feedback inhibition, which not only instantiates attractor states for forming categorical choices but also long transients for gradually accumulating evidence in favor of or against alternative options. Such a circuit endowed with reward-dependent synaptic plasticity is able to produce adaptive choice behavior. While decision threshold is a core concept for reaction time tasks, it can be dissociated from a general decision rule. Moreover, perceptual decisions and value-based economic choices are described within a unified framework in which probabilistic choices result from irregular neuronal activity as well as iterative interactions of a decision maker with an uncertain environment or other unpredictable decision makers in a social group.     

Spike timing precision of neuronal circuits

Spike timing is believed to be a key factor in sensory information encoding and computations performed by the neurons and neuronal circuits. However, the considerable noise and variability, arising from the inherently stochastic mechanisms that exist in the neurons and the synapses, degrade spike timing precision. Computational modeling can help decipher the mechanisms utilized by the neuronal circuits in order to regulate timing precision. In this paper, we utilize semi-analytical techniques, which were adapted from previously developed methods for electronic circuits, for the stochastic characterization of neuronal circuits. These techniques, which are orders of magnitude faster than traditional Monte Carlo type simulations, can be used to directly compute the spike timing jitter variance, power spectral densities, correlation functions, and other stochastic characterizations of neuronal circuit operation. We consider three distinct neuronal circuit motifs: Feedback inhibition, synaptic integration, and synaptic coupling. First, we show that both the spike timing precision and the energy efficiency of a spiking neuron are improved with feedback inhibition. We unveil the underlying mechanism through which this is achieved. Then, we demonstrate that a neuron can improve on the timing precision of its synaptic inputs, coming from multiple sources, via synaptic integration: The phase of the output spikes of the integrator neuron has the same variance as that of the sample average of the phases of its inputs. Finally, we reveal that weak synaptic coupling among neurons, in a fully connected network, enables them to behave like a single neuron with a larger membrane area, resulting in an improvement in the timing precision through cooperation.     

Adult neurogenesis and functional plasticity in neuronal circuits

The adult brain is a plastic place. To ensure that the mature nervous system's control of behaviour is flexible in the face of a varying environment, morphological and physiological changes are possible at many levels, including that of the entire cell. In two areas of the adult brain - the olfactory bulb and the dentate gyrus - new neurons are generated throughout life and form an integral part of the normal functional circuitry. This process is not fixed, but highly modulated, revealing a plastic mechanism by which the brain's performance can be optimized for a given environment. The functional benefits of this whole-cell plasticity, however, remain a matter for debate.     

Robustness and regularity of oscillations in neuronal populations

We study a biologically plausible but computationally simplified integrate-and-fire neuronal model. Oscillatory activity is analyzed in the networks with and without self-connections. We perform a detailed scan of four major parameters that represent the properties of neurons and synapses: connection ratio, connection strengths, post-synaptic potential decay rate and soma's potential decay rate. It is observed that networks with different properties exhibit different periods and different patterns of synchrony. We find that generally these oscillations are robust against changes of parameters, meanwhile we also locate the parametric boundaries where oscillations break down.     

Emergent synchronous bursting of oxytocin neuronal network

When young suckle, they are rewarded intermittently with a let-down of milk that results from reflex secretion of the hormone oxytocin; without oxytocin, newly born young will die unless they are fostered. Oxytocin is made by magnocellular hypothalamic neurons, and is secreted from their nerve endings in the pituitary in response to action potentials (spikes) that are generated in the cell bodies and which are propagated down their axons to the nerve endings. Normally, oxytocin cells discharge asynchronously at 1–3 spikes/s, but during suckling, every 5 min or so, each discharges a brief, intense burst of spikes that release a pulse of oxytocin into the circulation. This reflex was the first, and is perhaps the best, example of a physiological role for peptide-mediated communication within the brain: it is coordinated by the release of oxytocin from the dendrites of oxytocin cells; it can be facilitated by injection of tiny amounts of oxytocin into the hypothalamus, and it can be blocked by injection of tiny amounts of oxytocin antagonist. Here we show how synchronized bursting can arise in a neuronal network model that incorporates basic observations of the physiology of oxytocin cells. In our model, bursting is an emergent behaviour of a complex system, involving both positive and negative feedbacks, between many sparsely connected cells. The oxytocin cells are regulated by independent afferent inputs, but they interact by local release of oxytocin and endocannabinoids. Oxytocin released from the dendrites of these cells has a positive-feedback effect, while endocannabinoids have an inhibitory effect by suppressing the afferent input to the cells.     

Decoupling through synchrony in neuronal circuits with propagation delays

The level of synchronization in distributed systems is often controlled by the strength of the interactions between individual elements. In brain circuits the connection strengths between neurons are modified under the influence of spike-timing-dependent plasticity (STDP) rules. Here we show that when recurrent networks with conduction delays exhibit population bursts, STDP rules exert a strong decoupling force that desynchronizes activity. Conversely, when activity in the network is random, the same rules can have a coupling and synchronizing influence. The presence of these opposing forces promotes the self-organization of spontaneously active neuronal networks to a state at the border between randomness and synchrony. The decoupling force of STDP may be engaged by the synchronous bursts occurring in the hippocampus during slow-wave sleep, leading to the selective erasure of information from hippocampal circuits as memories are established in neocortical areas.     

Positive feedback loops sustain repeating bursts in neuronal circuits

Voluntary movements in animals are often episodic, with abrupt onset and termination. Elevated neuronal excitation is required to drive the neuronal circuits underlying such movements; however, the mechanisms that sustain this increased excitation are largely unknown. In the medicinal leech, an identified cascade of excitation has been traced from mechanosensory neurons to the swim oscillator circuit. Although this cascade explains the initiation of excitatory drive (and hence swim initiation), it cannot account for the prolonged excitation (10–100 s) that underlies swim episodes. We present results of physiological and theoretical investigations into the mechanisms that maintain swimming activity in the leech. Although intrasegmental mechanisms can prolong stimulus-evoked excitation for more than one second, maintained excitation and sustained swimming activity requires chains of several ganglia. Experimental and modeling studies suggest that mutually excitatory intersegmental interactions can drive bouts of swimming activity in leeches. Our model neuronal circuits, which incorporated mutually excitatory neurons whose activity was limited by impulse adaptation, also replicated the following major experimental findings: (1) swimming can be initiated and terminated by a single neuron, (2) swim duration decreases with experimental reduction in nerve cord length, and (3) swim duration decreases as the interval between swim episodes is reduced.     

Respiratory oscillations and heat evolution in synchronous cultures of Candida utilis

Synchronous cultures of the budding yeast Candida utilis prepared by continuous-flow size selection showed respiratory oscillations when the energy source was either glucose, acetate or glycerol. The period of the oscillations was about one-third of the cell cycle time (i.e. about 0.5 h). No fluctuations in heat evolution could be detected. In organisms growing with acetate or glycerol, the effects of cyanide, N,N'-dicyclohexylcarbodi-imide and carbonyl cyanide m-chlorophenylhydrazone (maximum inhibition of respiration at respiratory maxima, maximum uncoupling of energy conservation at respiratory minima) suggest that the control mechanism responsible for the oscillations is mitochondrial respiratory control in vivo. The effects of cyanide and N,N'-dicyclohexylcarbodi-imide on the respiration of cultures growing synchronously with glucose were different from those for cultures growing with the non-fermentable substrates; this suggests that the mitochondrial respiratory system interacts with the early reactions of glucose utilization.     

Systems-level modeling of neuronal circuits for leech swimming

This paper describes a mathematical model of the neuronal central pattern generator (CPG) that controls the rhythmic body motion of the swimming leech. The systems approach is employed to capture the neuronal dynamics essential for generating coordinated oscillations of cell membrane potentials by a simple CPG architecture with a minimal number of parameters. Based on input/output data from physiological experiments, dynamical components (neurons and synaptic interactions) are first modeled individually and then integrated into a chain of nonlinear oscillators to form a CPG. We show through numerical simulations that the values of a few parameters can be estimated within physiologically reasonable ranges to achieve good fit of the data with respect to the phase, amplitude, and period. This parameter estimation leads to predictions regarding the synaptic coupling strength and intrinsic period gradient along the nerve cord, the latter of which agrees qualitatively with experimental observations.     

Optical and genetic approaches toward understanding neuronal circuits in zebrafish

Optical and genetic tools are beginning to revolutionize thestudies of neuronal circuits. Neurons can now be labeled withconventional or genetically encoded indicators that allow theiractivity to be monitored during behavior in intact animals.Laser ablations and genetic inactivation offer ways to perturbactivity of specific cells to test their contributions to behavior.These approaches promise to speed progress in the understandingof vertebrate networks in genetic model systems such as miceand zebrafish. Here we review some of the progress in applyingthese tools, with an emphasis on our work to develop and applythese approaches in the zebrafish model.