Cascades of Periodic Solutions in a Neural Circuit With Delays and Slow-Fast Dynamics

We analyse periodic solutions in a system of four delayed differential equations forced by periodic inputs representing two competing neural populations connected with fast mutual excitation and slow delayed inhibition. The combination of mechanisms generates a rich dynamical structure that we are able to characterize using slow-fast dissection and a binary classification of states. We previously proved the existence conditions of all possible states 1:1 locked to the inputs and applied this analysis to the tracking of the rhythms perceived when listening to alternating sequences of low and high tones. Here we extend this analysis using analytical and computational tools by proving the existence a set of n:1 periodically locked states and their location in parameter space. Firstly we examine cycle skipping states and find that they accumulate in an infinite cascade of period-incrementing bifurcations with increasing periods for decreasing values of the local input strength. Secondly we analyse periodic solutions that alternate between 1:1 locked states that repeat after an integer multiple of the input period (swapping states). We show that such states accumulate in similar bifurcation cascades with decreasing values of the lateral input strength. We report a parameter-dependent scaling constant for the ratio of widths of successive regions in the cascades, which generalises across cycle skipping and swapping states. The periodic states reported here - emergent behaviours in the model - can be linked to known phenomena in auditory perception that are beyond the original scope of the model’s design.

Frequency-preference response in covalent modification cycles under substrate sequestration conditions

Covalent modification cycles (CMCs) are basic units of signaling systems and their properties are well understood. However, their behavior has been mostly characterized in situations where the substrate is in excess over the modifying enzymes. Experimental data on protein abundance suggest that the enzymes and their target proteins are present in comparable concentrations, leading to substrate sequestration by the enzymes. In this enzyme-in-excess regime, CMCs have been shown to exhibit signal termination, the ability of the product to return to a stationary value lower than its peak in response to constant stimulation, while this stimulation is still active, with possible implications for the ability of systems to adapt to environmental inputs. We characterize the conditions leading to signal termination in CMCs in the enzyme-in-excess regime. We also demonstrate that this behavior leads to a preferred frequency response (band-pass filters) when the cycle is subjected to periodic stimulation, whereas the literature reports that CMCs investigated so far behave as low-pass filters. We characterize the relationship between signal termination and the preferred frequency response to periodic inputs and we explore the dynamic mechanism underlying these phenomena. Finally, we describe how the behavior of CMCs is reflected in similar types of responses in the cascades of which they are part. Evidence of protein abundance in vivo shows that enzymes and substrates are present in comparable concentrations, thus suggesting that signal termination and frequency-preference response to periodic inputs are also important dynamic features of cell signaling systems, which have been overlooked.

Optimization of Methanol Synthesis under Forced Periodic Operation

Traditionally, methanol is produced in large amounts from synthesis gas with heterogeneous Cu/ZnO/Al2O3 catalysts under steady state conditions. In this paper, the potential of alternative forced periodic operation modes is studied using numerical optimization. The focus is a well-mixed isothermal reactor with two periodic inputs, namely, CO concentration in the feed and total feed flow rate. Exploiting a detailed kinetic model which also describes the dynamics of the catalyst, a sequential NLP optimization approach is applied to compare optimal steady state solutions with optimal periodic regimes. Periodic solutions are calculated using dynamic optimization with a periodicity constraint. The NLP optimization is embedded in a multi-objective optimization framework to optimize the process with respect to two objective functions and generate the corresponding Pareto fronts. The first objective is the methanol outlet flow rate. The second objective is the methanol yield based on the total carbon in the feed. Additional constraints arising from the complex methanol reaction and the practical limitations are introduced step by step. The results show that significant improvements for both objective functions are possible through periodic forcing of the two inputs considered here.

Frequency preference response in covalent modification cycles under substrate sequestration conditions

Covalent modification cycles (CMCs) are basic units of signaling systems and their properties are well understood. However, the behavior of such systems has been mostly characterized in situations where the substrate is in excess over the modifying enzymes. Experimental data on protein abundance suggest that the enzymes and their target proteins are present in comparable concentrations, leading to a different scenario in which the substrate is mostly sequestered by the enzymes. In this enzyme-in-excess regime, CMCs have been shown to exhibit signal termination, the ability of the product to return to a stationary value lower than the its peak in response to constant stimulation, while this stimulation is still active, with possible implications for the ability of systems to adapt to environmental inputs. We characterize the conditions leading to signal termination in CMCs in the enzyme-in-excess regime. We also demonstrate that this behavior leads to a preferred frequency response (band-pass filters) when the cycle is subjected to periodic stimulation, while the literature reports that CMCs investigated so far behave as low pass filters. We characterize the relationship between signal termination and the preferred frequency response to periodic inputs and we explore the dynamic mechanism underlying these phenomena. Finally, we describe how the behavior of CMCs is reflected in similar types of responses in the cascades of which they are part. Evidence of protein abundance in vivo shows that enzymes and substrates are present in comparable concentrations, thus suggesting that signal termination and frequency preference response to periodic inputs are also important dynamic features of cell signaling systems, which have been overlooked.

Infinite-dimensional Lur’e systems with almost periodic forcing

We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion.

How rivers incise to survive periodic inputs of immobile landslide-derived boulders

<p>Bedrock landsliding provides a strong negative feedback on bedrock river incision by causing long-lived burial events and hence hiatuses in downcutting.  Nevertheless, rivers in tectonically active settings carve deep canyons despite being periodically inundated with immobile boulders. How is this possible? In this contribution, we explore the processes through which rivers incise bedrock canyons within the Franciscan mélange in the actively uplifting California Coast Range. The Franciscan mélange is well known for its “melting ice cream topography” in which slow-moving landslides (“earthflows”) festoon the walls of river canyons and deliver car- to house-sized boulders to channels.  </p><p>Analysis of valley widths and river long profiles over ∼19  km of Alameda Creek (185  km<sup>2</sup> drainage area) and Arroyo Hondo (200  km<sup>2</sup> drainage area) in central California shows a very consistent picture in which earthflows that intersect these channels deposit immobile boulders that force tens of meters of gravel aggradation for kilometers upstream, leading to apparently long-lived sediment storage and channel burial at these sites. In contrast, over a ∼30  km section of the Eel River (5547  km<sup>2</sup> drainage area), there are no knickpoints or aggradation upstream of locations where earthflows impinge on its channel. Neither boulder supply nor transport capacity explains this difference. Rather, we find that the dramatically different sensitivity of the two locations to landslide blocking is linked to differences in channel width relative to typical seasonal displacements of landslides. The Eel River is ∼5 times wider than the largest annual seasonal displacement. In contrast, during wet winters, earthflows are capable of crossing and blocking the entire channel width of Arroyo Hondo and Alameda Creek. Hence, by virtue of having wide valley bottoms, larger rivers are more likely to simply flow around the toes of earthflows.  </p><p>For the smaller rivers in our study area that are chronically buried in landslide debris, our field observations provide evidence for two processes that may allow periodic bedrock river incision. Narrow channels in the Franciscan mélange that are buried in debris can incise epigenetic gorges around the margins of boulder jams during periods of earthflow dormancy when boulders are no longer input into channels.  Alternatively, during periods of earthflow dormancy, abrasion (and hence size reduction) of boulders in place from suspended sediment may ultimately render boulders mobile.  </p><p>Without explicit representation of these three processes, modeling the coupling of hillslope and channel evolution in this setting is not possible. </p><p><br><br></p>