We introduce a simple, easy to implement, and computationally efficient tropical convolutional neural network architecture that is robust against adversarial attacks. We exploit the tropical nature of piece-wise linear neural networks by embedding the data in the tropical projective torus. This can be accomplished with a single additional hidden layer called a tropical embedding layer, and can in principle be added to any neural network architecture. We study the geometry of the resulting decision boundary, and find that like adversarial training and various regularization techniques that have been proposed, adding the tropical embedding layer tends to increase the number of linear regions associated with the decision boundaries. Our numerical experiments show that our approach achieves state-of-the-art levels of adversarial robustness, while requiring much less computational time than adversarial training.

Although various hypotheses on population coding such as a grandmother cell hypothesis have been proposed in neuroscience, it remains unclear how many types of cells are actually required to trigger behavior. In order to completely resolve this issue, it is in principle necessary to measure the activity of all the relevant neurons in the brain. Nevertheless, previous studies have been limited to partial observations due to technical constraints, and strong conclusions could not be drawn because there is always the possibility that unmeasured cells play an important role. However, in recent years, technology has been developed that allows for the whole measurement of neural activities such as those related to olfaction in C. elegans and Drosophila, and it is hoped that the controversy on the population coding will be resolved. At the same time, it has become necessary to develop new data analysis methods that will enable stronger assertions based on whole measurements. Here we selected Drosophila melanogaster as a model organism, and analyzed data obtained by simultaneously measuring almost all the glomeruli (37 glomeruli) in the antennal lobe, which is the bottleneck through which all odor information passes. Taking advantage of the full-coverage antennal lobe recording with as many as 84 odor stimuli, we aimed at developing a novel algorithm to predict the behavioral responses to odors from the recorded activities of a minimum number of olfactory glomeruli. After demonstrating the validity of our algebra-based algorithm through benchmark simulated data, we estimated the number of indispensable glomeruli, looking for as sparse representation as possible. Our lower-bound estimate that at least 5 glomeruli are required to predict the drosophila behaviors under broad olfactory contexts imposed a limit on possible neural codes and, for example, rejected the grandmother cell hypothesis. In this way, our new method can reveal the degree of sparseness in neural codes.

Deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical projective torus. Here we propose tropical embedding to transform a vector in the tropical projective torus to a vector in the Euclidean space via the tropical metric. We introduce a tropical neural network where the first layer is a tropical embedding layer and the following layers are the same as the classical ones. We prove that a tropical neural network is a universal approximator and we derive a backpropagation rule for deep tropical neural networks. Then we provide TensorFlow 2 codes for implementing a tropical neural network in the same fashion as the classical one, where the weights initialization problem is considered according to the extreme value statistics. We apply our method to empirical data including sequences of hemagglutinin for influenza virus from New York. Finally we show that a tropical neural network can be interpreted as a generalization of a tropical logistic regression.

Much evidence from biological theory and empirical data indicates that, gene trees, phylogenetic trees reconstructed from different genes (loci), do not have to have exactly the same tree topologies. Such incongruence between gene trees might be caused by some "unusual" evolutionary events, such as meiotic sexual recombination in eukaryotes or horizontal transfers of genetic material in prokaryotes. However, most of the gene trees are constrained by the tree topology of the underlying species tree, that is, the phylogenetic tree depicting the evolutionary history of the set of species under consideration. In order to discover "outlying" gene trees which do not follow the "main distribution(s)" of trees, we propose to apply the "tropical metric" with the max-plus algebra from tropical geometry to a non-parametric estimation of gene trees over the space of phylogenetic trees. In this research we apply the "tropical metric," a well-defined metric over the space of phylogenetic trees under the max-plus algebra, to non-parametric estimation of gene trees distribution over the tree space. Kernel density estimator (KDE) is one of the most popular non-parametric estimation of a distribution from a given sample, and we propose an analogue of the classical KDE in the setting of tropical geometry with the tropical metric which measures the length of an intrinsic geodesic between trees over the tree space. We estimate the probability of an observed tree by empirical frequencies of nearby trees, with the level of influence determined by the tropical metric. Then, with simulated data generated from the multispecies coalescent model, we show that the non-parametric estimation of the gene tree distribution using the tropical metric performs better than one using the Billera-Holmes-Vogtmann (BHV) metric developed by Weyenberg et al. in terms of computational times and accuracy. We then apply it to Apicomplexa data.

Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperplane under the tropical metric with the max-plus algebra. In this paper, first we show generalization error bounds of tropical SVMs over the tropical projective torus. While the generalization error bounds attained via Vapnik-Chervonenkis (VC) dimensions in a distribution-free manner still depend on the dimension, we also show numerically and theoretically by extreme value statistics that the tropical SVMs for classifying data points from two Gaussian distributions as well as empirical data sets of different neuron types are fairly robust against the curse of dimensionality. Extreme value statistics also underlie the anomalous scaling behaviors of the tropical distance between random vectors with additional noise dimensions. Finally, we define tropical SVMs over a function space with the tropical metric.

We address the synchronization properties of two pulse-coupled resonate-and-fire neuron (RFN) models. The RFN model is a spiking neuron model that has second-order membrane dynamics with a threshold and a reset value. Due to such dynamics, the RFN model exhibits subthreshold oscillation of the membrane potential, and is sensitive to the timing of stimuli. So far the existence of anti-phase synchronization states and their stability in a system of two pulse-coupled RFN models have been reported. However, the effects of the reset value after firing on such synchronization states have not been considered. The reset value may affect the sensitivity to the input timing, leading to change in synchronization properties in the pulse-coupled RFN models. We newly found out-of-phase burst synchronization states and related bifurcation phenomena depending on the coupling strength in the system as the reset value was changed. Focusing on the symmetry of the system, we analyzed the stability of such phenomena by using a firing time difference map constructed from ID return maps with respect to firing time difference between two neurons. The analyses revealed the global stability of the out-of-phase synchronization states and the existence of the type I intermittency chaotic behavior.

We considered a gamma distribution of interspike intervals as a statistical model for neuronal spike generation. A gamma distribution is a natural extension of the Poisson process taking the effect of a refractory period into account. The model is specified by two parameters: a time-dependent firing rate and a shape parameter that characterizes spiking irregularities of individual neurons. Because the environment changes over time, observed data are generated from a model with a time-dependent firing rate, which is an unknown function. A statistical model with an unknown function is called a semiparametric model and is generally very difficult to solve. We used a novel method of estimating functions in information geometry to estimate the shape parameter without estimating the unknown function. We obtained an optimal estimating function analytically for the shape parameter independent of the functional form of the firing rate. This estimation is efficient without Fisher information loss and better than maximum likelihood estimation. We suggest a measure of spiking irregularity based on the estimating function, which may be useful for characterizing individual neurons in changing environments.

We considered a gamma distribution of interspike intervals as a statisti- cal model for neuronal spike generation. The model parameters consist of a time-dependent firing rate and a shape parameter that characterizes spiking irregularities of individual neurons. Because the environment changes with time, observed data are generated from the time-dependent firing rate, which is an unknown function. A statistical model with an unknown function is called a semiparametric model, which is one of the unsolved problem in statistics and is generally very difficult to solve. We used a novel method of estimating functions in information geometry to estimate the shape parameter without estimating the unknown function. We analytically obtained an optimal estimating function for the shape parameter independent of the functional form of the firing rate. This estimation is efficient without Fisher information loss and better than maximum likelihood estimation.