Combinatorial Code Research Articles

Population Coding of Natural Electrosensory Stimuli by Midbrain Neurons.

Natural stimuli display spatiotemporal characteristics that typically vary over orders of magnitude, and their encoding by sensory neurons remains poorly understood. We investigated population coding of highly heterogeneous natural electrocommunication stimuli in Apteronotus leptorhynchus of either sex. Neuronal activities were positively correlated with one another in the absence of stimulation, and correlation magnitude decayed with increasing distance between recording sites. Under stimulation, we found that correlations between trial-averaged neuronal responses (i.e., signal correlations) were positive and higher in magnitude for neurons located close to another, but that correlations between the trial-to-trial variability (i.e., noise correlations) were independent of physical distance. Overall, signal and noise correlations were independent of stimulus waveform as well as of one another. To investigate how neuronal populations encoded natural electrocommunication stimuli, we considered a nonlinear decoder for which the activities were combined. Decoding performance was best for a timescale of 6 ms, indicating that midbrain neurons transmit information via precise spike timing. A simple summation of neuronal activities (equally weighted sum) revealed that noise correlations limited decoding performance by introducing redundancy. Using an evolution algorithm to optimize performance when considering instead unequally weighted sums of neuronal activities revealed much greater performance values, indicating that midbrain neuron populations transmit information that reliably enable discrimination between different stimulus waveforms. Interestingly, we found that different weight combinations gave rise to similar discriminability, suggesting robustness. Our results have important implications for understanding how natural stimuli are integrated by downstream brain areas to give rise to behavioral responses.SIGNIFICANCE STATEMENT We show that midbrain electrosensory neurons display correlations between their activities and that these can significantly impact performance of decoders. While noise correlations limited discrimination performance by introducing redundancy, considering unequally weighted sums of neuronal activities gave rise to much improved performance and mitigated the deleterious effects of noise correlations. Further analysis revealed that increased discriminability was achieved by making trial-averaged responses more separable, as well as by reducing trial-to-trial variability by eliminating noise correlations. We further found that multiple combinations of weights could give rise to similar discrimination performances, which suggests that such combinatorial codes could be achieved in the brain. We conclude that the activities of midbrain neuronal populations can be used to reliably discriminate between highly heterogeneous stimulus waveforms.

Expansion and Accelerated Evolution of 9-Exon Odorant Receptors in Polistes Paper Wasps.

Independent origins of sociality in bees and ants are associated with independent expansions of particular odorant receptor (OR) gene subfamilies. In ants, one clade within the OR gene family, the 9-exon subfamily, has dramatically expanded. These receptors detect cuticular hydrocarbons (CHCs), key social signaling molecules in insects. It is unclear to what extent 9-exon OR subfamily expansion is associated with the independent evolution of sociality across Hymenoptera, warranting studies of taxa with independently derived social behavior. Here, we describe OR gene family evolution in the northern paper wasp, Polistes fuscatus, and compare it to four additional paper wasp species spanning ∼40 million years of evolutionary divergence. We find 200 putatively functional OR genes in P. fuscatus, matching predictions from neuroanatomy, and more than half of these are in the 9-exon subfamily. Most OR gene expansions are tandemly arrayed at orthologous loci in Polistes genomes, and microsynteny analysis shows species-specific gain and loss of 9-exon ORs within tandem arrays. There is evidence of episodic positive diversifying selection shaping ORs in expanded subfamilies. Values of omega (dN/dS) are higher among 9-exon ORs compared to other OR subfamilies. Within the Polistes OR gene tree, branches in the 9-exon OR clade experience relaxed negative (relaxed purifying) selection relative to other branches in the tree. Patterns of OR evolution within Polistes are consistent with 9-exon OR function in CHC perception by combinatorial coding, with both natural selection and neutral drift contributing to interspecies differences in gene copy number and sequence.

Expeditious Extraction of Histones from Limited Cells or Tissue Samples and Quantitative Top-Down Proteomic Analysis.

Histones are the primary protein component of chromatin and are involved in virtually all DNA-templated processes. Histones are abundantly post-translationally modified by a variety of chromatin-modifying machinery. These post-translational modifications (PTMs) are recognized by a range of "reader" proteins, which recruit additional proteins to specific locations on chromatin and impart precise and powerful effects on gene regulation. Each PTM typically exerts a positive or negative effect on transcription, and recent studies have shown that histone PTMs function in a combinatorial histone code: that is, histone PTMs function in combination to exert precise DNA-templated regulation. Thus, there is a need to identify and understand proteoforms, or unambiguously defined single protein molecules with all combinations of modifications. Top-down proteomics is currently the only viable approach for identifying and quantitating histone proteoforms, and mass spectrometry instruments have become sufficiently powerful to perform these quantitative analyses in a robust and high-throughput fashion. These recent innovations have enabled new experimental directions in chromatin research but have also introduced temporal and other constraints. This has led us to develop the protocols described here, which increase throughput, reduce sample requirements, and maintain robust quantitation. Although originally designed for high-throughput quantitative top-down proteomics, the protocols described here are useful for a wide range of chromatin biology applications. Starting with small amounts of cells or tissue, we describe two basic protocols for exceptionally rapid and efficient nuclei isolation, acid extraction of histones, and high-performance liquid chromatography fractionation of histones into histone families. We additionally describe the quantitative top-down proteomic analysis of histone H4 proteoforms. © 2021 Wiley Periodicals LLC. Basic Protocol 1: Nuclei isolation and acid extraction of histones from mammalian cells in culture/tissues Basic Protocol 2: HPLC fractionation of histones and histone H4 HPLC-MS/MS Support Protocol: Preparation of intact H3 histone tails by Glu-C digestion.

Olfactory bulb granule cells: specialized to link coactive glomerular columns for percept generation and discrimination of odors

The role of granule cells in olfactory processing is surrounded by several enigmatic observations, such as the purpose of reciprocal spines and the mechanisms for GABA release, the apparently low firing activity and recurrent inhibitory drive of granule cells, the missing proof for functional reciprocal connectivity, and the apparently negligible contribution to lateral inhibition. Here, we summarize recent results with regard to both the mechanisms of GABA release and the behavioral relevance of granule cell activity during odor discrimination. We outline a novel hypothesis that has the potential to resolve most of these enigmas and allows further predictions on the function of granule cells in odor processing. Briefly, recent findings imply that GABA release from the reciprocal spine requires a local spine action potential and the cooperative action of NMDA receptors and high voltage-activated Ca2+ channels. Thus, lateral inhibition is conditional on activity in the principal neurons connected to a granule cell and tightly intertwined with recurrent inhibition. This notion allows us to infer that lateral inhibition between principal neurons occurs “on demand,” i.e., selectively on coactive mitral and tufted cells, and thus can provide directed, dynamically switched lateral inhibition in a sensory system with 1000 input channels organized in glomerular columns. The mechanistic underpinnings of this hypothesis concur with findings from odor discrimination behavior in mice with synaptic proteins deleted in granule cells. In summary, our hypothesis explains the unusual microcircuit of the granule cell reciprocal spine as a means of olfactory combinatorial coding.

Interspike intervals within retinal spike bursts combinatorially encode multiple stimulus features.

Neurons in various regions of the brain generate spike bursts. While the number of spikes within a burst has been shown to carry information, information coding by interspike intervals (ISIs) is less well understood. In particular, a burst with k spikes has k-1 intraburst ISIs, and these k-1 ISIs could theoretically encode k-1 independent values. In this study, we demonstrate that such combinatorial coding occurs for retinal bursts. By recording ganglion cell spikes from isolated salamander retinae, we found that intraburst ISIs encode oscillatory light sequences that are much faster than the light intensity modulation encoded by the number of spikes. When a burst has three spikes, the two intraburst ISIs combinatorially encode the amplitude and phase of the oscillatory sequence. Analysis of trial-to-trial variability suggested that intraburst ISIs are regulated by two independent mechanisms responding to orthogonal oscillatory components, one of which is common to bursts with a different number of spikes. Therefore, the retina encodes multiple stimulus features by exploiting all degrees of freedom of burst spike patterns, i.e., the spike number and multiple intraburst ISIs.

Lower Bounds for Total Storage of Multiset Combinatorial Batch Codes Using Linear Programming

The class of multiset combinatorial batch codes (MCBCs) was introduced by Zhang et al. (2018) as a generalization of combinatorial batch codes (CBCs), which are replication-based batch codes. The MCBCs allow multiple users to retrieve items in parallel in a distributed storage and a fundamental objective in this study is to determine the minimum total storage given certain requirements. We formulate linear programs so that the optimal solutions provide lower bounds on the total storage of MCBCs. Borrowing techniques from linear programming, we improve known lower bounds in some cases. Furthermore, for some parameters, we showed that these lower bounds are either tight or asymptotically tight by constructing the corresponding codes.

Encoding the Odor of Cigarette Smoke.

The encoding of odors is believed to begin as a combinatorial code consisting of distinct patterns of responses from odorant receptors (ORs), trace-amine associated receptors (TAARs), or both. To determine how specific response patterns arise requires detecting patterns in vivo and understanding how the components of an odor, which are nearly always mixtures of odorants, give rise to parts of the pattern. Cigarette smoke, a common and clinically relevant odor consisting of >400 odorants, evokes responses from 144 ORs and 3 TAARs in freely behaving male and female mice, the first example of in vivo responses of both ORs and TAARs to an odor. As expected, a simplified artificial mimic of cigarette smoke odor tested at low concentration to identify highly sensitive receptors evokes responses from four ORs, all also responsive to cigarette smoke. Human subjects of either sex identify 1-pentanethiol as the odorant most critical for perception of the artificial mimic; and in mice the OR response patterns to these two odors are significantly similar. Fifty-eight ORs respond to the headspace above 25% 1-pentanethiol, including 9 ORs responsive to cigarette smoke. The response patterns to both cigarette smoke and 1-pentanethiol have strongly responsive ORs spread widely across OR sequence diversity, consistent with most other combinatorial codes previously measured in vivo The encoding of cigarette smoke is accomplished by a broad receptor response pattern, and 1-pentanethiol is responsible for a small subset of the responsive ORs in this combinatorial code.SIGNIFICANCE STATEMENT Complex odors are usually perceived as distinct odor objects. Cigarette smoke is the first complex odor whose in vivo receptor response pattern has been measured. It is also the first pattern shown to include responses from both odorant receptors and trace-amine associated receptors, confirming that the encoding of complex odors can be enriched by signals coming through both families of receptors. Measures of human perception and mouse receptor physiology agree that 1-pentanethiol is a critical component of a simplified odorant mixture designed to mimic cigarette smoke odor. Its receptor response pattern helps to link those of the artificial mimic and real cigarette smoke, consistent with expectations about perceptual similarity arising from shared elements in receptor response patterns.

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length n as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in O(log2⁡n) amortized time per word, while the best generation algorithm hitherto has O(n) running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages.

Remarks on external contextual grammars with selection

In an external contextual grammar with selection, contexts are associated with selection languages, and the context (u,v) can be wrapped around a word w if and only if w belongs to the selection language associated with (u,v). In the papers [1], [3], [4], and [6], some hierarchies are presented if selection languages are taken from subregular language families, defined by automata-theoretic and combinatorial properties, codes, and ideals. In this paper we merge and complete the hierarchies.

Valence and State-Dependent Population Coding in Dopaminergic Neurons in the Fly Mushroom Body

Neuromodulation permits flexibility of synapses, neural circuits, and ultimately behavior. One neuromodulator, dopamine, has been studied extensively in its role as a reward signal during learning and memory across animal species. Newer evidence suggests that dopaminergic neurons (DANs) can modulate sensory perception acutely, thereby allowing an animal to adapt its behavior and decision making to its internal and behavioral state. In addition, some data indicate that DANs are not homogeneous but rather convey different types of information as a heterogeneous population. We have investigated DAN population activity and how it could encode relevant information about sensory stimuli and state by taking advantage of the confined anatomy of DANs innervating the mushroom body (MB) of the fly Drosophila melanogaster. Using invivo calcium imaging and a custom 3D image registration method, we found that the activity of the population of MB DANs encodes innate valence information of an odor or taste as well as the physiological state of the animal. Furthermore, DAN population activity is strongly correlated with movement, consistent with a role of dopamine in conveying behavioral state to the MB. Altogether, our data and analysis suggest that DAN population activities encode innate odor and taste valence, movement, and physiological state in a MB-compartment-specific manner. We propose that dopamine shapes innate perception through combinatorial population coding of sensory valence, physiological, and behavioral context.

Molecular design of hypothalamus development.

A wealth of specialized neuroendocrine command systems intercalated within the hypothalamus control the most fundamental physiological needs1,2. Nevertheless, a developmental blueprint integrating molecular determinants of neuronal and glial diversity along temporal and spatial scales of hypothalamus development remains unresolved3. Here, we combine single-cell RNA-seq on 51,199 cells of ectodermal origin, gene regulatory network (GRN) screens in conjunction with GWAS-based disease phenotyping and genetic lineage reconstruction to show that 9 glial and 33 neuronal subtypes are generated by mid-gestation under the control of distinct GRNs. Combinatorial molecular codes arising from neurotransmitters, neuropeptides and transcription factors are minimally required to decode the taxonomical hierarchy of hypothalamic neurons. Differentiation of GABA and dopamine but not glutamate neurons relies on quasi-stable intermediate states with a pool of GABA progenitors giving rise to dopamine cells4. An unexpected abundance of chemotropic proliferation and guidance cues commonly implicated in dorsal (cortical) patterning5 was found in the hypothalamus. Particularly, Slit/Robo loss-of-function impacted both the production and positioning of periventricular dopamine neurons. Overall, we uncover molecular principles shaping the developmental architecture of the hypothalamus and show how neuronal heterogeneity is transformed into a multimodal neural unit to endow a virtually infinite adaptive potential throughout life.

Neural Circuit Dynamics for Sensory Detection.

We consider the question of how sensory networks enable the detection of sensory stimuli in a combinatorial coding space. We are specifically interested in the olfactory system, wherein recent experimental studies have reported the existence of rich, enigmatic response patterns associated with stimulus onset and offset. This study aims to identify the functional relevance of such response patterns (i.e., what benefits does such neural activity provide in the context of detecting stimuli in a natural environment). We study this problem through the lens of normative, optimization-based modeling. Here, we define the notion of a low-dimensional latent representation of stimulus identity, which is generated through action of the sensory network. The objective of our optimization framework is to ensure high-fidelity tracking of a nominal representation in this latent space in an energy-efficient manner. It turns out that the optimal motifs emerging from this framework possess morphologic similarity with prototypical onset and offset responses observed in vivo in locusts (Schistocerca americana) of either sex. Furthermore, this objective can be exactly achieved by a network with reciprocal excitatory-inhibitory competitive dynamics, similar to interactions between projection neurons and local neurons in the early olfactory system of insects. The derived model also makes several predictions regarding maintenance of robust latent representations in the presence of confounding background information and trade-offs between the energy of sensory activity and resultant behavioral measures such as speed and accuracy of stimulus detection.SIGNIFICANCE STATEMENT A key area of study in olfactory coding involves understanding the transformation from high-dimensional sensory stimulus to low-dimensional decoded representation. Here, we examine not only the dimensionality reduction of this mapping but also its temporal dynamics, with specific focus on stimuli that are temporally continuous. Through optimization-based synthesis, we examine how sensory networks can track representations without prior assumption of discrete trial structure. We show that such tracking can be achieved by canonical network architectures and dynamics, and that the resulting responses resemble observations from neurons in the insect olfactory system. Thus, our results provide hypotheses regarding the functional role of olfactory circuit activity at both single neuronal and population scales.

Alternatively Constructed Estrogen Receptor Alpha-Driven Super-Enhancers Result in Similar Gene Expression in Breast and Endometrial Cell Lines.

Super-enhancers (SEs) are clusters of highly active enhancers, regulating cell type-specific and disease-related genes, including oncogenes. The individual regulatory regions within SEs might be simultaneously bound by different transcription factors (TFs) and co-regulators, which together establish a chromatin environment conducting to effective transcription. While cells with distinct TF profiles can have different functions, how different cells control overlapping genetic programs remains a question. In this paper, we show that the construction of estrogen receptor alpha-driven SEs is tissue-specific, both collaborating TFs and the active SE components greatly differ between human breast cancer-derived MCF-7 and endometrial cancer-derived Ishikawa cells; nonetheless, SEs common to both cell lines have similar transcriptional outputs. These results delineate that despite the existence of a combinatorial code allowing alternative SE construction, a single master regulator might be able to determine the overall activity of SEs.

Modulation of the combinatorial code of odorant receptor response patterns in odorant mixtures

The perception of odors relies on combinatorial codes consisting of odorant receptor (OR) response patterns to encode odor identity. Modulation of these patterns by odorant interactions at ORs potentially explains several olfactory phenomena: mixture suppression, unpredictable sensory outcomes, and the perception of odorant mixtures as unique objects. We determined OR response patterns to 4 odorants and 3 binary mixtures in vivo in mice, identifying 30 responsive ORs. These patterns typically had a few strongly responsive ORs and a greater number of weakly responsive ORs. ORs responsive to an odorant were often unrelated sequences distributed across several OR subfamilies. Mixture responses predicted pharmacological interactions between odorants, which were tested in vitro by heterologous expression of ORs in cultured cells, providing independent evidence confirming odorant agonists for 13 ORs and identifying both suppressive and additive effects. This included 11 instances of antagonism of ORs by an odorant, 1 instance of additive responses to a binary mixture, 1 instance of suppression of a strong agonist by a weak agonist, and the discovery of an inverse agonist for an OR. Interactions between odorants at ORs are common even when the odorants are not known to interact perceptually in humans, and in some cases interactions at mouse ORs correlate with the ability of humans to perceive an odorant in a mixture.

Optimality of sparse olfactory representations is not affected by network plasticity.

The neural representation of a stimulus is repeatedly transformed as it moves from the sensory periphery to deeper layers of the nervous system. Sparsening transformations are thought to increase the separation between similar representations, encode stimuli with great specificity, maximize storage capacity of associative memories, and provide an energy efficient instantiation of information in neural circuits. In the insect olfactory system, odors are initially represented in the periphery as a combinatorial code with relatively simple temporal dynamics. Subsequently, in the antennal lobe this representation is transformed into a dense and complex spatiotemporal activity pattern. Next, in the mushroom body Kenyon cells (KCs), the representation is dramatically sparsened. Finally, in mushroom body output neurons (MBONs), the representation takes on a new dense spatiotemporal format. Here, we develop a computational model to simulate this chain of olfactory processing from the receptor neurons to MBONs. We demonstrate that representations of similar odorants are maximally separated, measured by the distance between the corresponding MBON activity vectors, when KC responses are sparse. Sparseness is maintained across variations in odor concentration by adjusting the feedback inhibition that KCs receive from an inhibitory neuron, the Giant GABAergic neuron. Different odor concentrations require different strength and timing of feedback inhibition for optimal processing. Importantly, as observed in vivo, the KC–MBON synapse is highly plastic, and, therefore, changes in synaptic strength after learning can change the balance of excitation and inhibition, potentially leading to changes in the distance between MBON activity vectors of two odorants for the same level of KC population sparseness. Thus, what is an optimal degree of sparseness before odor learning, could be rendered sub–optimal post learning. Here, we show, however, that synaptic weight changes caused by spike timing dependent plasticity increase the distance between the odor representations from the perspective of MBONs. A level of sparseness that was optimal before learning remains optimal post-learning.

Morphisms of Neural Codes

We define a notion of morphism between combinatorial codes, making the class of all combinatorial codes into a category $\mathbf{Code}$. We show that morphisms can be used to remove redundant information from a code, and that morphisms preserve convexity. This fact leads us to define "minimally non-convex" codes. We propose a program to characterize these minimal obstructions to convexity and hence characterize all convex codes. We implement a library of Sage code to perform computation with morphisms. These computational methods yield the smallest to-date example of a non-convex code with no local obstructions. We conclude by giving an algebraic formulation of our results.

ПІДВИЩЕННЯ ЕФЕКТИВНОСТІ ЦИКЛІЧНИХ КОДІВ МЕТОДАМИ КОМБІНАТОРНОЇ ОПТИМІЗАЦІЇ

The methods of improving the cyclic codes efficiency constructed on the basis of combinatorial configurations of the type "ideal ring bundles" (IRB) s by three factors – correction ability, power of coding method and complexity of the decoding procedure are considered. The method is based on the principle of combinatorial optimization, grounded on the algebraic theory of ordered integer sequences with a circular structure, all the numbers, as well as all sums of consecutive numbers exhaust the value sofnatural row numbers. Two theoretically grounded approaches to increase of noise immunity of cyclic codes are offered: implementation of optimized IRB-code, as well as monolithic and group one. Optimized cyclic IRB-code favorably differs from the rest of the codes of this class by the highest correction capacity at the same length of code words. Optimized IRB-codes constitute a large group of cyclic codes designed on a combinatorial models with selection of corresponding relationships between the parameters of the code to achieve its specified technical characteristics. Noise protected monolithic and group codes belong to the group of self-correcting codes with a ring structure and probabilistic assessment of the level of noise protection. This property allow so instant lydetect a particular part or all invalid characters in the code word by the majority principle. Mathematical calculations have been performed to calculate the optimized ratios between the parameters of cyclic IRB-codes, under which they reach maximum correction capacity. The algorithm of constructing and increasing the power of coding methods of optimized noise-resistant IRB-codes is examined and analyzed. The concrete examples of increase efficiency of combinatorial optimization cyclic codes methods with appropriate calculations and tables are given. The comparative analysis of the IRB-codes with the Golay codes and Bose – Chaudhuri – Hocquenghe (BCH) codes with respect to correction ability, power encoding method and computational complexity of decoding procedures is carried out. The advantages and disadvantages of cyclic, and ringmonolithic and group IRB-codes in comparison with classical analogues are determined. The prospect so fusing the research results in the problems of information and communication technologies are outlined.

Hybridization Chain Reaction for Quantitative and Multiplex Imaging of Gene Expression in Amphioxus Embryos and Adult Tissues.

In situ hybridization (ISH) methods remain the most popular approach for profiling the expression of a gene at high spatial resolution and have been broadly used to address many biological questions. One compelling application is in the field of evo-devo, where comparing gene expression patterns has offered insight into how vertebrate development has evolved. Gene expression profiling in the invertebrate chordate amphioxus (cephalochordate) has been particularly instrumental in this context: its key phylogenetic position as sister group to all other chordates makes it an ideal model system to compare with vertebrates and for reconstructing the ancestral condition of our phylum. However, while ISH methods have been developed extensively in vertebrate model systems to fluorescently detect the expression of multiple genes simultaneously at a cellular and subcellular resolution, amphioxus gene expression profiling is still based on single-gene nonfluorescent chromogenic methods, whose spatial resolution is often compromised by diffusion of the chromogenic product. This represents a serious limitation for reconciling gene expression dynamics between amphioxus and vertebrates and for molecularly identifying cell types, defined by their combinatorial code of gene expression, that may have played pivotal roles in evolutionary innovation. Herein we overcome these problems by describing a new protocol for application of the third-generation hybridization chain reaction (HCR) to the amphioxus, which permits fluorescent, multiplex, and quantitative detection of gene expression in situ, within the changing morphology of the developing embryo, and in adult tissues. A detailed protocol is herein provided for whole-mount preparations of embryos and vibratome sections of adult tissues.

A combinatorial code of transcription factors specifies subtypes of visual motion-sensing neurons in Drosophila.

ABSTRACTDirection-selective T4/T5 neurons exist in four subtypes, each tuned to visual motion along one of the four cardinal directions. Along with their directional tuning, neurons of each T4/T5 subtype orient their dendrites and project their axons in a subtype-specific manner. Directional tuning, thus, appears strictly linked to morphology in T4/T5 neurons. How the four T4/T5 subtypes acquire their distinct morphologies during development remains largely unknown. Here, we investigated when and how the dendrites of the four T4/T5 subtypes acquire their specific orientations, and profiled the transcriptomes of all T4/T5 neurons during this process. This revealed a simple and stable combinatorial code of transcription factors defining the four T4/T5 subtypes during their development. Changing the combination of transcription factors of specific T4/T5 subtypes resulted in predictable and complete conversions of subtype-specific properties, i.e. dendrite orientation and matching axon projection pattern. Therefore, a combinatorial code of transcription factors coordinates the development of dendrite and axon morphologies to generate anatomical specializations that differentiate subtypes of T4/T5 motion-sensing neurons.

Algebraic models, inverse problems, and pseudomonomials from biology

In contrast to the extensive use of linear algebra in biology, problems involving nonlinear polynomials—the field of algebraic biology—are more conspicuous. In this article, we highlight two biological problems where similar algebraic structures arise in very different contexts. Specifically, we will look at algebraic models of molecular networks, and combinatorial codes of place fields in neuroscience. Both of these topics involve inverse problems where the data is encoded with pseudomonomials, simple algebraic objects that do not seem to have been studied much on their own, but have gained considerable attention lately from their visibility in mathematical biology. Though this article is algebraic in nature, it is written for the general mathematician with a minimal algebra background assumed. It provides two distinctive features that have not yet appeared in the literature: (i) a survey of Boolean and logical modeling from a computational algebra perspective, and (ii) a unification of this topic with algebraic neuroscience by highlighting the role of pseudomonomials in both fields.