The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

When walking along a city street, you might encounter a range of scents and odors, from the smells of coffee and food to those of exhaust fumes and garbage. The odors are swept to your nose by air currents that move scents in two different ways. They carry them downwind in a process called advection, but they also mix them chaotically with clean air in a process called turbulence. What results is an odor plume: a complex ever-changing structure resembling the smoke rising from a chimney. Within a plume, areas of highly concentrated odor particles break up into smaller parcels as they travel further from the odor source. This means that the concentration of the odor does not vary along a smooth gradient. Instead, the odor arrives in brief and unpredictable bursts. Despite this complexity, insects are able to use odor plumes with remarkable ease to navigate towards food sources. But how do they do this? Answering this question has proved challenging because odor plumes are usually invisible. Over the years, scientists have come up with a number of creative solutions to this problem, including releasing soap bubbles together with odors, or using wind tunnels to generate simpler, straight plumes in known locations. These approaches have shown that when insects encounter an odor, they surge upwind towards its source. When they lose track of the odor, they cast themselves crosswind in an effort to regain contact. But this does not explain how insects are able to navigate irregular odor plumes, in which both the timing and location of the odor bursts are unpredictable. Demir, Kadakia et al. have now bridged this gap by showing how fruit flies are attracted to smoke, an odorant that is also visible. By injecting irregular smoke plumes into a custom-built wind tunnel, and then imaging flies as they walked through it, Demir, Kadakia et al. showed that flies make random halts when navigating the plume. Each time they stop, they use the timing of the odor bursts reaching them to decide when to start moving again. Rather than turning every time they detect an odor, flies initiate turns at random times. When several odor bursts arrive in a short time, the flies tend to orient these turns upwind rather than downwind. Flies therefore rely on a different strategy to navigate irregular odor plumes than the ‘surge and cast’ method they use for regular odor streams. Successful navigation through complex irregular plumes involves a degree of random behavior. This helps the flies gather information about an unpredictable environment as they search for the source of the odor. These findings may help to understand how other insects use odor to navigate in the real world, for example, how mosquitoes track down human hosts.

Walking Drosophila navigate complex plumes using stochastic decisions biased by the timing of odor encounters

Odour plumes in the wild are spatially complex and rapidly fluctuating structures carried by turbulent airflows1–4. To successfully navigate plumes in search of food and mates, insects must extract and integrate multiple features of the odour signal, including odour identity5, intensity6 and timing6–12. Effective navigation requires balancing these multiple streams of olfactory information and integrating them with other sensory inputs, including mechanosensory and visual cues9,12,13. Studies dating back a century have indicated that, of these many sensory inputs, the wind provides the main directional cue in turbulent plumes, leading to the longstanding model of insect odour navigation as odour-elicited upwind motion6,8–12,14,15. Here we show that Drosophila melanogaster shape their navigational decisions using an additional directional cue—the direction of motion of odours—which they detect using temporal correlations in the odour signal between their two antennae. Using a high-resolution virtual-reality paradigm to deliver spatiotemporally complex fictive odours to freely walking flies, we demonstrate that such odour-direction sensing involves algorithms analogous to those in visual-direction sensing16. Combining simulations, theory and experiments, we show that odour motion contains valuable directional information that is absent from the airflow alone, and that both Drosophila and virtual agents are aided by that information in navigating naturalistic plumes. The generality of our findings suggests that odour-direction sensing may exist throughout the animal kingdom and could improve olfactory robot navigation in uncertain environments. Odour motion contains valuable directional information that is absent from the airflow alone, and Drosophila use this directional information to shape their navigational decisions. 

Odour motion sensing enhances navigation of complex plumes

We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously, and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involve coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time dependent and time independent coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time dependent coupling function u(t) we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time dependent coupling appears to have benefits in synchronizing long datasets with minimal impact, while the initial value method with time independent coupling tends to be substantially faster, more flexible and easier to use. We also describe a new method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time dependent injected current.

Parameter and State Estimation of Experimental Chaotic Systems Using Synchronization

This tutorial illustrates the use of data assimilation algorithms to estimate unobserved variables and unknown parameters of conductance-based neuronal models. Modern data assimilation (DA) techniques are widely used in climate science and weather prediction, but have only recently begun to be applied in neuroscience. The two main classes of DA techniques are sequential methods and variational methods. We provide computer code implementing basic versions of a method from each class, the Unscented Kalman Filter and 4D-Var, and demonstrate how to use these algorithms to infer several parameters of the Morris–Lecar model from a single voltage trace. Depending on parameters, the Morris–Lecar model exhibits qualitatively different types of neuronal excitability due to changes in the underlying bifurcation structure. We show that when presented with voltage traces from each of the various excitability regimes, the DA methods can identify parameter sets that produce the correct bifurcation structure even with initial parameter guesses that correspond to a different excitability regime. This demonstrates the ability of DA techniques to perform nonlinear state and parameter estimation and introduces the geometric structure of inferred models as a novel qualitative measure of estimation success. We conclude by discussing extensions of these DA algorithms that have appeared in the neuroscience literature.

/pdf/data-assimilation-methods-for-neuronal-state-and-parameter-38xegie737.pdf

Data Assimilation Methods for Neuronal State and Parameter Estimation

Networks of nonlinear systems contain unknown parameters and dynamical degrees of freedom that may not be observable with existing instruments. From observable state variables, we want to estimate the connectivity of a model of such a network and determine the full state of the model at the termination of a temporal observation window during which measurements transfer information to a model of the network. The model state at the termination of a measurement window acts as an initial condition for predicting the future behavior of the network. This allows the validation (or invalidation) of the model as a representation of the dynamical processes producing the observations. Once the model has been tested against new data, it may be utilized as a predictor of responses to innovative stimuli or forcing. We describe a general framework for the tasks involved in the “inverse” problem of determining properties of a model built to represent measured output from physical, biological, or other processes when the ...

A unifying view of synchronization for data assimilation in complex nonlinear networks.

We showed previously (Gorur-Shandilya et al., 2017) that Drosophila olfactory receptor neurons (ORNs) expressing the co-receptor Orco scale their gain inversely with mean odor intensity according to Weber-Fechner's law. Here, we show that this front-end adaptation promotes the reconstruction of odor identity from dynamic odor signals, even in the presence of confounding background odors and rapid intensity fluctuations. These enhancements are further aided by known downstream transformations in the antennal lobe and mushroom body. Our results, which are applicable to various odor classification and reconstruction schemes, stem from the fact that this adaptation mechanism is not intrinsic to the identity of the receptor involved. Instead, a feedback mechanism adjusts receptor sensitivity based on the activity of the receptor-Orco complex, according to Weber-Fechner's law. Thus, a common scaling of the gain across Orco-expressing ORNs may be a key feature of ORN adaptation that helps preserve combinatorial odor codes in naturalistic landscapes.

Front-end Weber-Fechner gain control enhances the fidelity of combinatorial odor coding.

Odor identity is encoded by spatiotemporal patterns of activity in olfactory receptor neurons (ORNs). In natural environments, the intensity and timescales of odor signals can span several orders of magnitude, and odors can mix with one another, potentially scrambling the combinatorial code mapping neural activity to odor identity. Recent studies have shown that in Drosophila the ORNs that express the olfactory co-receptor Orco scale their gain inversely with mean odor concentration according to the Weber-Fechner Law of psychophysics. Here we use a minimal biophysical model of signal transduction, ORN firing, and signal decoding to investigate the implications of this front-end scaling law for the neural representations of odor identity. We find that Weber-Fechner scaling enhances coding capacity and promotes the reconstruction of odor identity from dynamic odor signals, even in the presence of confounding background odors and rapid intensity fluctuations. We show that these enhancements are further aided by downstream transformations such as normalization and divergence in the antennal lobe and mushroom body. Thus, despite the broad overlap between individual ORN tuning curves, a mechanism of front-end adaptation, when endowed with Weber-Fechner scaling, may play a vital role in preserving representations of odor identity in naturalistic odor landscapes.

https://biorxiv.org/content/biorxiv/early/2018/11/21/475103.full-text.pdf

Front-end Weber-Fechner gain control enhances the fidelity of combinatorial odor coding

We and others have shown that during odor plume navigation, walking Drosophila melanogaster bias their motion upwind in response to both the frequency of their encounters with the odor (Demir et al., 2020), and the intermittency of the odor signal, i.e. the fraction of time the signal is above a detection threshold (Alvarez-Salvado et al., 2018). Here we combine and simplify previous mathematical models that recapitulated these data to investigate the benefits of sensing both of these temporal features, and how these benefits depend on the spatiotemporal statistics of the odor plume. Through agent-based simulations, we find that navigators that only use frequency or intermittency perform well in some environments - achieving maximal performance when gains are near those inferred from experiment - but fail in others. Robust performance across diverse environments requires both temporal modalities. However, we also find a steep tradeoff when using both sensors simultaneously, suggesting a strong benefit to modulating how much each sensor is weighted, rather than using both in a fixed combination across plumes.

/pdf/navigating-a-diversity-of-turbulent-plumes-is-enhanced-by-28luie92zb.pdf

Navigating a diversity of turbulent plumes is enhanced by sensing complementary temporal features of odor signals

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