More Like this

1. Root System Chip-Firing II: Central-Firing

   https://doi.org/10.1093/imrn/rnz112  

   Galashin, Pavel; Hopkins, Sam; McConville, Thomas; Postnikov, Alexander (June 2019, International Mathematics Research Notices)
2. Root system chip-firing

   Galashin, Pavel; Hopkins, Sam; McConville, Thomas; Postnikov, Alexander (October 2018, Séminaire lotharingien de combinatoire)

   Propp recently introduced a variant of chip-firing on the infinite path where the chips are given distinct integer labels and conjectured that this process is confluent from certain (but not all) initial configurations of chips. Hopkins, McConville, and Propp proved Propp's confluence conjecture. We recast this result in terms of root systems: the labeled chip-firing game can be seen as a process which allows replacing an integer vector λ by λ+α whenever λ is orthogonal to α, for α a positive root of a root system of Type A. We give conjectures about confluence for this process in the general setting of an arbitrary root system. We show that the process is always confluent from any initial point after modding out by the action of the Weyl group (an analog of unlabeled chip-firing in arbitrary type). We also study some remarkable deformations of this process which are confluent from any initial point. For these deformations, the set of weights with given stabilization has an interesting geometric structure related to permutohedra. This geometric structure leads us to define certain `Ehrhart-like' polynomials that conjecturally have nonnegative integer coefficients. 

   more » « less
3. Chip-Firing Games and Critical Groups

   https://doi.org/10.1007/978-3-030-37853-0_4  

   Glass, G; Kaplan, N. (April 2020, Foundations for undergraduate research in mathematics)

   null; null; null (Ed.)
4. Fluid Dynamics of Ballistic Strategies in Nematocyst Firing

   https://doi.org/10.3390/fluids5010020  

   Hamlet, Christina; Strychalski, Wanda; Miller, Laura (March 2020, Fluids)

   Nematocysts are stinging organelles used by members of the phylum Cnidaria (e.g., jellyfish, anemones, hydrozoans) for a variety of important functions including capturing prey and defense. Nematocysts are the fastest-known accelerating structures in the animal world. The small scale (microns) coupled with rapid acceleration (in excess of 5 million g) present significant challenges in imaging that prevent detailed descriptions of their kinematics. The immersed boundary method was used to numerically simulate the dynamics of a barb-like structure accelerating a short distance across Reynolds numbers ranging from 0.9–900 towards a passive elastic target in two dimensions. Results indicate that acceleration followed by coasting at lower Reynolds numbers is not sufficient for a nematocyst to reach its target. The nematocyst’s barb-like projectile requires high accelerations in order to transition to the inertial regime and overcome the viscous damping effects normally encountered at small cellular scales. The longer the barb is in the inertial regime, the higher the final velocity of the projectile when it touches its target. We find the size of the target prey does not dramatically affect the barb’s approach for large enough values of the Reynolds number, however longer barbs are able to accelerate a larger amount of surrounding fluid, which in turn allows the barb to remain in the inertial regime for a longer period of time. Since the final velocity is proportional to the force available for piercing the membrane of the prey, high accelerations that allow the system to persist in the inertial regime have implications for the nematocyst’s ability to puncture surfaces such as cellular membranes or even crustacean cuticle. 

   more » « less
5. Neuronal Firing Rate as Code Length: A Hypothesis

   Qian, Ning; Zhang, Jun (January 2019, Computational brain & behavior)

   Many theories assume that a sensory neuron’s higher firing rate indicates a greater probability of its preferred stimulus. However, this contradicts 1) the adaptation phenomena where prolonged exposure to, and thus increased probability of, a stimulus reduces the firing rates of cells tuned to the stimulus; and 2) the observation that unexpected (low probability) stimuli capture attention and increase neuronal firing. Other theories posit that the brain builds predictive/efficient codes for reconstructing sensory inputs. However, they cannot explain that the brain preserves some information while discarding other. We propose that in sensory areas, projection neurons’ firing rates are proportional to optimal code length (i.e., negative log estimated probability), and their spike patterns are the code, for useful features in inputs. This hypothesis explains adaptation-induced changes of V1 orientation tuning curves, and bottom-up attention. We discuss how the modern minimum-description-length (MDL) principle may help understand neural codes. Because regularity extraction is relative to a model class (defined by cells) via its optimal universal code (OUC), MDL matches the brain’s purposeful, hierarchical processing without input reconstruction. Such processing enables input compression/understanding even when model classes do not contain true models. Top-down attention modifies lower-level OUCs via feedback connections to enhance transmission of behaviorally relevant information. Although OUCs concern lossless data compression, we suggest possible extensions to lossy, prefix-free neural codes for prompt, online processing of most important aspects of stimuli while minimizing behaviorally relevant distortion. Finally, we discuss how neural networks might learn MDL’s normalized maximum likelihood (NML) distributions from input data. 

   more » « less