Gating of homeostatic regulation of intrinsic excitability produces cryptic long-term storage of prior perturbations.

Neurons and neuronal circuits must maintain their function throughout the life of the organism despite changing environments. Previous theoretical and experimental work suggests that neurons monitor their activity using intracellular calcium concentrations to regulate their intrinsic excitability. Models with multiple sensors can distinguish among different patterns of activity, but previous work using models with multiple sensors produced instabilities that lead the models' conductances to oscillate and then to grow without bound and diverge. We now introduce a nonlinear degradation term that explicitly prevents the maximal conductances to grow beyond a bound. We combine the sensors' signals into a master feedback signal that can be used to modulate the timescale of conductance evolution. Effectively, this means that the negative feedback can be gated on and off according to how far the neuron is from its target. The modified model recovers from multiple perturbations. Interestingly, depolarizing the models to the same membrane potential with current injection or with simulated high extracellular K+ produces different changes in conductances, arguing that caution must be used in interpreting manipulations that serve as a proxy for increased neuronal activity. Finally, these models accrue traces of prior perturbations that are not visible in their control activity after perturbation but that shape their responses to subsequent perturbations. These cryptic or hidden changes may provide insight into disorders such as posttraumatic stress disorder that only become visible in response to specific perturbations.

Repeated applications of high potassium elicit long-term changes in a motor circuit from the crab, Cancer borealis.

We examined the effects of altered extracellular potassium concentration on the output of the well-studied pyloric circuit in the crab, Cancer borealis. Pyloric neurons initially become quiescent, then recover spiking and bursting activity in high potassium saline (2.5x[K+]). These changes in circuit robustness are maintained after the perturbation is removed; pyloric neurons are more robust to subsequent potassium perturbations even after several hours of wash in control saline. Despite this long-term "memory" of the stimulus history, we found no differences in neuronal activity in control saline. The circuit's adaptation is erased by both low potassium saline (0.4x[K+]) and direct hyperpolarizing current. Initial sensitivity of PD neurons to high potassium saline also varies seasonally, indicating that changes in robustness may reflect natural changes in circuit states. Thus, perturbation, followed by recovery of normal activity, can hide cryptic changes in neuronal properties that are only revealed by subsequent challenges.

Temperature compensation in a small rhythmic circuit.

Temperature affects the conductances and kinetics of the ionic channels that underlie neuronal activity. Each membrane conductance has a different characteristic temperature sensitivity, which raises the question of how neurons and neuronal circuits can operate robustly over wide temperature ranges. To address this, we employed computational models of the pyloric network of crabs and lobsters. We produced multiple different models that exhibit a triphasic pyloric rhythm over a range of temperatures and explored the dynamics of their currents and how they change with temperature. Temperature can produce smooth changes in the relative contributions of the currents to neural activity so that neurons and networks undergo graceful transitions in the mechanisms that give rise to their activity patterns. Moreover, responses of the models to deletions of a current can be different at high and low temperatures, indicating that even a well-defined genetic or pharmacological manipulation may produce qualitatively distinct effects depending on the temperature.

Author Correction: Single-trial classification of awareness state during anesthesia by measuring critical dynamics of global brain activity.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Single-trial classification of awareness state during anesthesia by measuring critical dynamics of global brain activity.

In daily life, in the operating room and in the laboratory, the operational way to assess wakefulness and consciousness is through responsiveness. A number of studies suggest that the awake, conscious state is not the default behavior of an assembly of neurons, but rather a very special state of activity that has to be actively maintained and curated to support its functional properties. Thus responsiveness is a feature that requires active maintenance, such as a homeostatic mechanism to balance excitation and inhibition. In this work we developed a method for monitoring such maintenance processes, focusing on a specific signature of their behavior derived from the theory of dynamical systems: stability analysis of dynamical modes. When such mechanisms are at work, their modes of activity are at marginal stability, neither damped (stable) nor exponentially growing (unstable) but rather hovering in between. We have previously shown that, conversely, under induction of anesthesia those modes become more stable and thus less responsive, then reversed upon emergence to wakefulness. We take advantage of this effect to build a single-trial classifier which detects whether a subject is awake or unconscious achieving high performance. We show that our approach can be developed into a means for intra-operative monitoring of the depth of anesthesia, an application of fundamental importance to modern clinical practice.

Visualization of currents in neural models with similar behavior and different conductance densities.

Conductance-based models of neural activity produce large amounts of data that can be hard to visualize and interpret. We introduce visualization methods to display the dynamics of the ionic currents and to display the models' response to perturbations. To visualize the currents' dynamics, we compute the percent contribution of each current and display them over time using stacked-area plots. The waveform of the membrane potential and the contribution of each current change as the models are perturbed. To represent these changes over a range of the perturbation control parameter, we compute and display the distributions of these waveforms. We illustrate these procedures in six examples of bursting model neurons with similar activity but that differ as much as threefold in their conductance densities. These visualization methods provide heuristic insight into why individual neurons or networks with similar behavior can respond widely differently to perturbations.

Complex spatiotemporal behavior and coherent excitations in critically-coupled chains of neural circuits.

We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the "synaptic matrix" M embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a "critical" synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period τ , thus given by a unitary matrix U = exp ⁡ ( τ 2 M ) . It is now explicit that such critical couplings are volume-preserving in the sense of Liouville's theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.

Complex behavior in chains of nonlinear oscillators.

This article outlines sufficient conditions under which a one-dimensional chain of identical nonlinear oscillators can display complex spatio-temporal behavior. The units are described by phase equations and consist of excitable oscillators. The interactions are local and the network is poised to a critical state by balancing excitation and inhibition locally. The results presented here suggest that in networks composed of many oscillatory units with local interactions, excitability together with balanced interactions is sufficient to give rise to complex emergent features. For values of the parameters where complex behavior occurs, the system also displays a high-dimensional bifurcation where an exponentially large number of equilibria are borne in pairs out of multiple saddle-node bifurcations.

Nonlinear resonances and multi-stability in simple neural circuits.

This article describes a numerical procedure designed to tune the parameters of periodically driven dynamical systems to a state in which they exhibit rich dynamical behavior. This is achieved by maximizing the diversity of subharmonic solutions available to the system within a range of the parameters that define the driving. The procedure is applied to a problem of interest in computational neuroscience: a circuit composed of two interacting populations of neurons under external periodic forcing. Depending on the parameters that define the circuit, such as the weights of the connections between the populations, the response of the circuit to the driving can be strikingly rich and diverse. The procedure is employed to find circuits that, when driven by external input, exhibit multiple stable patterns of periodic activity organized in complex tuning diagrams and signatures of low dimensional chaos.

Loss of Consciousness Is Associated with Stabilization of Cortical Activity.

What aspects of neuronal activity distinguish the conscious from the unconscious brain? This has been a subject of intense interest and debate since the early days of neurophysiology. However, as any practicing anesthesiologist can attest, it is currently not possible to reliably distinguish a conscious state from an unconscious one on the basis of brain activity. Here we approach this problem from the perspective of dynamical systems theory. We argue that the brain, as a dynamical system, is self-regulated at the boundary between stable and unstable regimes, allowing it in particular to maintain high susceptibility to stimuli. To test this hypothesis, we performed stability analysis of high-density electrocorticography recordings covering an entire cerebral hemisphere in monkeys during reversible loss of consciousness. We show that, during loss of consciousness, the number of eigenmodes at the edge of instability decreases smoothly, independently of the type of anesthetic and specific features of brain activity. The eigenmodes drift back toward the unstable line during recovery of consciousness. Furthermore, we show that stability is an emergent phenomenon dependent on the correlations among activity in different cortical regions rather than signals taken in isolation. These findings support the conclusion that dynamics at the edge of instability are essential for maintaining consciousness and provide a novel and principled measure that distinguishes between the conscious and the unconscious brain. SIGNIFICANCE STATEMENT: What distinguishes brain activity during consciousness from that observed during unconsciousness? Answering this question has proven difficult because neither consciousness nor lack thereof have universal signatures in terms of most specific features of brain activity. For instance, different anesthetics induce different patterns of brain activity. We demonstrate that loss of consciousness is universally and reliably associated with stabilization of cortical dynamics regardless of the specific activity characteristics. To give an analogy, our analysis suggests that loss of consciousness is akin to depressing the damper pedal on the piano, which makes the sounds dissipate quicker regardless of the specific melody being played. This approach may prove useful in detecting consciousness on the basis of brain activity under anesthesia and other settings.

Parameter estimation, nonlinearity, and Occam's razor.

Nonlinear systems are capable of displaying complex behavior even if this is the result of a small number of interacting time scales. A widely studied case is when complex dynamics emerges out of a nonlinear system being forced by a simple harmonic function. In order to identify if a recorded time series is the result of a nonlinear system responding to a simpler forcing, we develop a discrete nonlinear transformation for time series based on synchronization techniques. This allows a parameter estimation procedure which simultaneously searches for a good fit of the recorded data, and small complexity of a fluctuating driving parameter. We illustrate this procedure using data from respiratory patterns during birdsong production.

Dynamical criticality during induction of anesthesia in human ECoG recordings.

In this work we analyze electro-corticography (ECoG) recordings in human subjects during induction of anesthesia with propofol. We hypothesize that the decrease in responsiveness that defines the anesthetized state is concomitant with the stabilization of neuronal dynamics. To test this hypothesis, we performed a moving vector autoregressive analysis and quantified stability of neuronal dynamics using eigenmode decomposition of the autoregressive matrices, independently fitted to short sliding temporal windows. Consistent with the hypothesis we show that while the subject is awake, many modes of neuronal activity oscillations are found at the edge of instability. As the subject becomes anesthetized, we observe statistically significant increase in the stability of neuronal dynamics, most prominently observed for high frequency oscillations. Stabilization was not observed in phase randomized surrogates constructed to preserve the spectral signatures of each channel of neuronal activity. Thus, stability analysis offers a novel way of quantifying changes in neuronal activity that characterize loss of consciousness induced by general anesthetics.

Temperature induced syllable breaking unveils nonlinearly interacting timescales in birdsong motor pathway.

The nature of telencephalic control over premotor and motor circuits is debated. Hypotheses range from complete usurping of downstream circuitry to highly interactive mechanisms of control. We show theoretically and experimentally, that telencephalic song motor control in canaries is consistent with a highly interactive strategy. As predicted from a theoretical model of respiratory control, mild cooling of a forebrain nucleus (HVC) led to song stretching, but further cooling caused progressive restructuring of song, consistent with the hypothesis that respiratory gestures are subharmonic responses to a timescale present in the output of HVC. This interaction between a life-sustaining motor function (respiration) and telencephalic song motor control suggests a more general mechanism of how nonlinear integration of evolutionarily new brain structures into existing circuitry gives rise to diverse, new behavior.

Average dynamics of a driven set of globally coupled excitable units.

We investigate the behavior of the order parameter describing the collective dynamics of a large set of driven, globally coupled excitable units. We derive conditions on the parameters of the system that allow to bound the degree of synchrony of its solutions. We describe a regime where time dependent nonsynchronous dynamics occurs and, yet, the average activity displays low dimensional, temporally complex behavior.

Low-dimensional dynamical model for the diversity of pressure patterns used in canary song.

During song production, oscine birds produce large air sac pressure pulses. During those pulses, energy is transferred to labia located at the juncture between the bronchii and the trachea, inducing the high frequency labial oscillations which are responsible for airflow modulations, i.e., the uttered sound. In order to generate diverse syllables, canaries (Serinus canaria) use a set of air sac pressure patterns with characteristic shapes. In this work we show that these different shapes can be approximated by the subharmonic solutions of a forced normal form. This simple model is built from identifying dynamical elements which allow to reproduce the shape of the pressure pattern corresponding to one syllable type. Remarkably, integrating that simple model for other parameters allows to recover the other pressure patterns used during song. Interpreting the diversity of these physiological gestures as subharmonic solutions of a simple nonlinear system allows us to account simultaneously for their morphological features as well as for the syllabic timing and suggests a strategy for the generation of complex motor patterns.