Manicka S, Pai VP, Levin M.

             embryo development
             paracrine signaling
             embryonic brain development

	Graphical abstract
	Figure 1. Schematic summary of the endogenous voltage prepattern’s control of embryonic brain patterning in Xenopus and an overview of the neural plate circuit model of this process (A) Summary of our previous studies80,87,88,97 showing that the difference (dashed black line) in voltage across the hyperpolarized neural plate and relatively depolarized surrounding ectoderm is essential for driving correct gene expression pattern, large-scale brain morphology, and normal learning behavior. (B) Endogenous embryonic voltage prepattern resulting in normal brain morphology.80,87,88,97 (C) Embryonic manipulation of ion fluxes, exposure to teratogens (nicotine and ethanol), or genetic/biochemical disruption (Notch disruption) leading to depolarization of the neural plate erases the critical voltage difference between neural plate and ectoderm, resulting in mispatterned gene expression, and defects in large-scale brain morphology and learning behavior.80,87,88,97,105,106 (D) Embryonic manipulation of ion fluxes leading to hyperpolarization of surrounding ectoderm also erases the critical voltage difference between neural plate and ectoderm, resulting in mispatterned gene expression and abnormal large-scale brain morphology.80,97 (E) Both local (neural plate) and distant (ectodermal) interventions (manipulating ion fluxes by channel misexpression or drugs targeting ion channels) that restore this critical voltage difference between neural plate and ectoderm, restore gene expression, large-scale brain morphology, and learning behavior.80,87,88,97,105,106 (F) A schematic of our model. The overt structure of the model is a 2-dimensional lattice approximately representing the neural plate tissue (shown in the illustration). The circles indicate the individual cells of the tissue, with the colors indicating different voltages corresponding to the endogenous voltage prepattern (blue means depolarized and red, hyperpolarized). Each cell possesses a generic hyperpolarizing channel and depolarizing channel (not shown) and a generic gene regulatory network (gray networks enclosed in squares). The cells exchange gene products and proteins with neighboring cells through the IN-generic intercellular connections (thick gray lines with bidirectional arrowheads) representing paracrine signaling pathways. These collectively represent a plethora of paracrine signaling pathways such as calcium, cAMP, serotonin, integrin, Notch, etc. In this current first iteration model, we generalize them all under paracrine signals as the model is consistent with them all and does not depend on their precise nature. Of note, the IN does not directly allow the exchange of ions but it does so only indirectly via the voltage→gene→gene→voltage pathway (Figure 2A). The voltage and the gene network of a cell influence each other via dedicated connections (thin black lines with unidirectional arrowheads): the voltage acts as an external input to a subset of the genes, affecting their expression and, in turn, the expressions of a subset of the genes can dynamically alter the conductance of the ion channels. The cells in the model are symmetrical, that is, they do not possess a preferred polarity such as that imposed by planar cell polarity. The 4x6 dimension of the lattice shown here is the size that was used during training. (G) Schematic of the input-output map of the pattern discrimination problem; the three input patterns on the left correspond to the embryo schematics displayed in C, B, and D, respectively, with the depolarized (red) and hyperpolarized (blue) voltages corresponding to about −5.2 mV (−10 mV) and −52 mV (−55 mV) in the model (empirical observations80).
	Figure 2. Overview of the mathematical details of the neural plate circuit model and the process of identifying its parameters (training) (A) The equations defining the model, with the variables and parameters mapped to a simplified two-cell version (the rest of the model is symmetrical to this version). The parameters (red) are learned using machine-learning techniques. The bioelectric constants, namely, , , , and , are described in ref.88 and in Tables S1–S8. (B) The model is trained using a combination of genetic algorithm and gradient descent. The final model referred to throughout this paper is a product of a training loop that starts with a population of randomly parametrized models, simulates all of them, and then modifies them using genetic algorithm and gradient descent to improve their ability to solve the pattern discrimination problem.
	Figure 3. Architecture of the chosen high-performing model Depicted here is the connectivity of a two-cell version of the full model; the rest of the network is symmetrical to this version (full model parameters described in Tables S1–S8). For clarity, we show only about 20% of the strongest connections (10% of the most positive in purple and 10% of the most negative in orange/brown); edge weights are linearly normalized to the range (−1, 1). The most prominent connections run from the voltage to gene 5 (purple), with a positive weight, and among genes 3 and 5 both within a cell and between cells (brown), with a negative weight. Connections running from genes to voltage do exist in the full model, but they were filtered out due to their weak weights.
	Figure 4. The model not only discriminates between the correct and the incorrect voltage patterns by mapping them to distinct gene expression values but also recapitulates empirically observed development of the voltage pattern (A) Timeseries of the mean normalized gene expression, averaged over all cells and genes, for the three different input conditions depicted on the right: “Depolarized” (all cells are initially depolarized), “Endogenous” (the left and right two columns are initially depolarized with the middle two columns hyperpolarized), or “Hyperpolarized” (all cells are initially hyperpolarized). Insets: (left) discrimination scores of the six individual genes. The red dashed horizontal line indicates the discrimination score expected from a randomly parametrized model. (Right) Gene expression timeseries for gene 6, the gene with the highest discrimination score; the scales of the axes match the scales of the main plot. More detailed timeseries resolved at the level of cells and genes are presented in Figure S2. (B) Top: comparison of the modeled and empirical spatial voltage profiles of a horizontal cross-section of the embryo (schematic) at a point in time when the voltages corresponding to the three different input conditions (treatments) have fully developed (∼200th timestep in the model). Bottom: the spatiotemporal voltage profile of the model depicting the timeseries of Vmem for each cell under each of the three input conditions. Each square in the 4x6 layout of the tissue refers to an individual cell.
	Figure 5. The best fit model flexibly scales to larger tissues when solving the voltage pattern discrimination problem The timeseries shows the mean expression, averaged over all cells, of the discriminator gene in a scaled up 10x18 model, initialized with the endogenous voltage pattern (inset; left and right six columns are depolarized, and the middle six columns are hyperpolarized), showing that it successfully discriminates between the correct and incorrect voltage patterns. More detailed timeseries resolved at the cell of the level and for larger tissues are presented in Figure S3.
	Figure 6. Differential change in model performance due to knocking out individual or patches of cells in the tissue The darker the shade of a block, the higher the drop in performance due to knocking out that block. A cell or a patch was “knocked out” by freezing the states of its corresponding genes at zero and dropping their biases to negative infinity, thereby rendering them ineffectual both as inputs and outputs. (A) A 4x6 tissue where individual cells were knocked out. (B) A 10x18 tissue where 2-cell x 3-cell patches (each block is a patch) were knocked out.
	Figure 7. Analytical reconstruction of the gene expression timeseries using only the second-order influence of the voltage pattern on gene expression A comparison between the normalized original expression timeseries of the discriminator gene (dashed) for the endogenous input pattern and the corresponding reconstruction (solid) obtained from the second-order (Hessian) influence of the endogenous voltage pattern reveals their similarity. The timeseries was reconstructed using the equation , where refers to the set of cells in the top left quadrant of the tissue and was optimized to fit the observed normalized with the initial condition . Only the cells in the 2x3 top left quadrant of the full 4x6 tissue (dashed boxes) were considered because the model is bilaterally symmetrical both about the vertical and horizontal axes.
	Figure 8. Spatiotemporal dynamics of the second-order control of gene expression by the endogenous voltage pattern Each undirected edge represents a pair of cells whose voltages collectively influence the expression of an average discriminator gene in the top left quadrant. The weight of these edges indicates the degree of influence (second-order derivative) exerted by the voltages of the corresponding pairs of cells on the discriminator gene, whereas their colors indicate whether the influence is positive or negative (strong purple means strongly positive and strong orange/brown means strongly negative). These edges represent the summation term in the equation , where refers to the set of cells in the top left quadrant of the tissue and was optimized to fit the observed normalized with the initial condition . Only the cells in the 2x3 top left quadrant of the full 4x6 tissue (dashed boxes) were considered because the model is bilaterally symmetrical both about the vertical and horizontal axes. For clarity, only about 10% of the observed edges are shown. The arrows running from one band to another serve as a visual illustration of the average weighted and signed influence exerted by the three bands on the target top left quadrant of the pattern.
	Figure 9. A division of labor exists among the genes in terms of their sensitivities to the different spatial scales of the voltage pattern The difference between the first-order (Jacobian) directional derivatives of the gene activities computed with respect to the voltage at the tissue level and the single-cell level tends to be positive for genes 2 and 5, negative for genes 3 and 4, and mostly balanced but interspersed with positive spurts for genes 1 and 6. This quantity, known as the net scale sensitivity of a gene, is defined as . Here, the first term is a directional derivative and computes the net sensitivity to the tissue, and the second term refers to the single cell. The term is a normalization term for computing the directional derivative along the direction of the unit vector, thus enabling the comparison between the two scales of the tissue and the single cell. The corresponding inclinations of sensitivity are toward the tissue level, single-cell level, and an intermediate level, respectively. Due to the symmetry of the model, only the genes contained by the cells in the 2x3 top left quadrant of the full 4x6 tissue were considered for these calculations.
	Figure 10. Model predictions about the responses to two types of non-endogenous input voltage patterns with different polarization ratios simulated in a 10x18 tissue Average activity levels of the discriminator gene in response to (A) a voltage pattern with a less skewed ratio of polarization in comparison to the endogenous pattern: a half-and-half (step function) voltage pattern where the left half is hyperpolarized and the right half is depolarized (inset); (B) a voltage pattern with a more skewed ratio of polarization in comparison to the endogenous pattern: a sharpened voltage pattern where the overall pattern is maintained but the hyperpolarized neural plate (central band) is only a third of its original normal size (inset).
	Figure 11. Perturbation of voltage in one-half of an embryo, creating a step function voltage pattern, does not cause defects in brain morphology (A) Illustration of Xenopus embryo at stage 3 (four-cell), indicating two dorsal blastomeres as main precursors of eye and brain and two ventral blastomeres as main precursors of non-neural tissues.137,171 (B) Normal voltage pattern in control (uninjected) stage ∼15 Xenopus embryos. Top: illustration of expected membrane voltage distribution in stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of CC2-DMPE:DiBAC voltage-reporter-dyes-stained embryos, showing characteristic hyperpolarization in the neural plate (yellow arrows) and surrounding depolarized ectoderm80,87,88 (N = 6) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD. (C) Step function voltage pattern at stage ∼15 in embryos microinjected with Kv1.5 + β-galactosidase mRNA in one dorsal blastomere at four-cell stage. Top: illustration of expected membrane voltage distribution in injected stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of CC2-DMPE: DiBAC voltage-reporter-dyes-stained embryos, showing characteristic hyperpolarization in neural plate (yellow arrow) but hyperpolarized ectoderm only on the injected side (red arrow) (N = 6) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD. (D–G) Representative images of stage 45 tadpoles. (D, E) Tadpoles from uninjected (D) or injected (E) embryos. Blue arrowheads indicate intact nostrils, orange brackets indicate intact forebrain (FB), yellow brackets indicate intact midbrain (MB), cyan brackets indicate intact hindbrain (HB) and intact eyes (e) (F, G) β-Galactosidase expression assessed using X-Gal (blue) in bleached tadpoles. In injected tadpoles (G), X-gal staining is evident in the eye and brain on the injected side (blue arrowheads) but not the uninjected side (magenta arrowheads), validating our targeting of mRNA microinjection. There was no X-gal staining (magenta arrowheads) in uninjected tadpoles (F) (N > 10 tadpoles per experimental group). (H) Quantification of brain morphology defects in stage 45 tadpoles under different injection conditions demonstrates no significant differences among any of the injection conditions. All injections were made into the right dorsal blastomere at 4-cell stage as indicated in the schematic under each graph. Percentage of tadpoles with brain defects for each experimental group are as follows: Controls: 8%, β-galactosidase: 7%, Kv1.5 + β-galactosidase: 8%, and Kir4.1: 6%. Data are mean ± SD, n.s. = non-significant (one-way ANOVA with Tukey’s post-hoc test for n = 3 independent experiments with N > 50 embryos per treatment group per experiment).
	Figure 12. Perturbation of voltage to decrease neural plate hyperpolarized cells while minimally changing the overall voltage pattern leads to significant brain morphology defects (A) Illustration of Xenopus embryo at stage 3 (four-cell) indicating two dorsal blastomeres as main precursors of eye and brain and two ventral blastomeres as main precursors of non-neural tissues.137,171 (B) Endogenous voltage pattern in control (uninjected) stage ∼15 embryos. Top: illustration of expected membrane voltage distribution in stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of embryo stained with CC2-DMPE:DiBAC4 voltage reporter dyes, showing characteristic hyperpolarization in the neural plate (yellow arrows) and surrounding depolarized ectoderm80,87,88 (N = 6) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD. (C) Altered voltage pattern in stage ∼15 embryos microinjected with DN-KATP + β-galactosidase mRNA in one dorsal blastomere at four-cell stage. Top: illustration of expected membrane voltage distribution in injected stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of injected embryo stained with CC2-DMPE: DiBAC4 membrane voltage reporter dyes showing reduced region of hyperpolarization in neural plate and narrower hyperpolarization region (empty yellow arrow) (N > 6 embryos per experimental group) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD. (D–G) Representative images of stage 45 tadpoles. (D, E) Tadpoles from uninjected (D) or microinjected with DN-KATP + β-galactosidase mRNA (E) embryos. Blue arrowheads indicate intact nostrils; orange, yellow, and cyan brackets indicate intact forebrain (FB), midbrain (MB), and hindbrain (HB), respectively. (e) indicates intact eyes, and orange arrowheads indicate mispatterned brain and eye. (F, G) β-Galactosidase expression assessed using X-Gal (blue) in bleached tadpoles either left uninjected (controls) or co-injected with DN-KATP + β-galactosidase mRNA. There was no X-gal staining (magenta arrowheads) in uninjected tadpoles (F) In injected tadpoles (G), β-galactosidase was observed in the eye and brain on the injected side (blue arrowheads) but not the uninjected side (magenta arrowheads), confirming expected targeting of microinjected mRNAs. (N > 10 tadpoles per experimental group). (H) DN-KATP + β-galactosidase injection significantly increased the percentage of stage 45 tadpoles with brain morphology defects: uninjected or β-galactosidase controls—both 9%; DN-KATP + β-galactosidase—53%. All injections were made into the right dorsal blastomere at four-cell stage as indicated in the schematic under each graph. Data are mean ± SD, ∗∗p < 0.01, n.s. = non-significant (one-way ANOVA with Tukey’s post-hoc test for n = 3 independent experiments with N > 50 embryos per treatment group per experiment).
	Figure 13. An overview of the current knowledge and gaps in our understanding of Xenopus brain development (A) A schematic of the general state of our knowledge in embryonic brain patterning and the area of focus of this study. (B) A schematic of the functional relationship in which a spatial pattern (the characteristic bell curve voltage distribution, left panel) is transduced into specific gene expression decisions (middle panel, taken from with permission from ref.84) that is required to implement mature brain anatomy (right panel).
	Figure S1. Training improves performance. (A) GA search resulted in a gradual improvement of the performance of the bestperforming individual among the population up until about 1100 generations at which point one of the top two best-performing models was randomly chosen and further refined using GD. (B) Comparison of the performances of an ensemble of models before and after training. The after-training group is significantly better than the before-training group as per t-test (p<1e-6). The red star indicates the performance of the model analyzed in the main text. Related to Figs. 2,3.
	Figure S2. Cell-wise behavior of the individual genes while solving the pattern discrimination problem in a 4x6 tissue. Most genes, regardless of the cell, tend to activate (positive values) for the endogenous bioelectric pattern while tending to deactivate (negative values) for the aberrant patterns. It can also be noticed that while the quantitative gene expression behavior depends on the cell’s spatial location (up to bilateral symmetry across both the vertical and horizontal axes passing through the center of the tissue), there are no significant qualitative differences. Related to Fig. 4.
	Figure S3. Behavior of the discriminator gene in tissues larger than see during training. (A) 10x18 tissue. Regardless of the cell, it can be seen that the discriminator gene tends to activate (positive values) for the endogenous bioelectric pattern (top) while tending to deactivate (negative values) for the aberrant patterns. It can also be noticed that while the quantitative gene expression behavior depends on the cell’s spatial location (corner cells tend to have lower activation levels, for example), there are no significant qualitative differences. (B) Top: 20 x 21 tissue (420 cells); Bottom: 20 x 30 tissue (600 cells). For simplicity, the average of the first 3 columns, the middle 24 columns and the last 3 columns are shown. It can be seen that the genes in the marginal columns tend to deactivate even for endogenous input bioelectric patterns. Related to Fig. 5.
	Figure S4. Cell-wise behavior of the discriminator gene while responding to aberrant bioelectric input patterns: (A) A half-andhalf Vmem pattern where the left half is hyperpolarized, and the right half is depolarized (inset) results in normal brain development as indicated by the activation of the discriminator gene in all cells, compared to deactivation for the uniform patterns. (B) A sharpened endogenous Vmem pattern where only a third of the hyperpolarized central band of the endogenous pattern is hyperpolarized (bottom; the middle two columns of the six at the center of this 10x18 tissue) results in an intermediate brain development due to the failure of gene expression to discriminate between the endogenous and the incorrect Vmem patterns in the leftmost and the rightmost two columns. Related to Fig. 10.
	Figure S5. The time-evolution of context-dependency of the individual cells in a 4x6 tissue while solving the pattern discrimination problem. The color of each cell indicates the extent to which the connections of that cell in the corresponding Hessian network originate from another depolarized cell in the left or the right bands or a hyperpolarized cell in the central band. Related to Fig. 8.
	Figure S6. A nonlinear recurrent dynamical system can effectively act like a linear feed-forward-like system when the forcing parameter controls the variable more strongly than the variable does over itself. Shown here are reconstructions of the original observed timeseries (Eqn. 5) using the optimized PTPE approximation (Eqn. 9), where (left) the parameter is small (� = 1); and (right) the parameter is relatively large (� = 10). In the latter case, the parameter effectively canalizes the dynamics of the variable. This is because a canalized system reduces the nonlinearity of the system by partly linearizing it – an effect of the leveraging of timeless Taylor coefficients. Related to Fig. 7.
	Figure S7. The quality of the simplified Taylor reconstructions with time-independent coefficients (PTPE) gets better with more canalization (larger values of the forcing parameter �). The reconstruction errors were computed as the mean squared error between the reconstructed and the original timeseries. Related to Fig. 7.
	Figure S8. The distributions of reconstruction errors, plotted on a log scale, for a suite of 100 randomly parametrized neural plate models obtained by individually optimizing the coefficients of the Jacobian and the Hessian terms. Red stars indicate the corresponding errors for the trained neural plate model described and analyzed in the main text. These plots show that though good fits may be expected with just the Jacobian (5-percentile MSE as low as 0.07), the Jacobian reconstruction MSE of our neural plate model is about 0.14 and is well placed within the expected range of 5 and 95 percentiles. The Hessian reconstruction errors, on the other hand, are higher than the corresponding Jacobian errors (5-percentile MSE is about 4.8). Although, the Hessian error of our neural plate, with an MSE of 0.23, is well below the expected range. These observations suggest that the neural plate model was optimized to operate at the second-order level. Related to Fig. 7.
	Figure S9. Gene-wise Hessian reconstruction of timeseries. A comparison of the normalized original gene expression timeseries of all the genes (dashed) and the corresponding reconstruction (solid) from the second order (Hessian) influence of the Vmem. The resemblance between the reconstruction and the original indicates that most of the gene activity is determined by its second-order relationship with Vmem. This moreover also suggests a canalization-like mechanism where the dynamics of this complex recurrent model is simplified in such a way that there is an almost direct control of the discriminator gene by the Vmem, as if the paths through the other genes are bypassed; this canalized pathway cannot be inferred from the structure of the static connectivity (Fig 3) alone. This makes intuitive sense since discriminatory behavior requires a comparison of Vmem of different cells, which the Hessian presumably aides by containing the differential information. Due to the symmetry of the model, only the genes contained by the cells in the 2x3 top left quadrant of the full 4x6 tissue (dashed boxes in Fig 9) were considered for these calculations; the Hessian depicted here is the sum of all Hessians corresponding to each cell in the quadrant. Related to Fig. 7.
	Figure S10. The discriminator gene dynamically processes the three resting potential pattern bands (Figure 1F) in a more complex way (A) compared to the average gene (B). The band-wise first order (Jacobian) normalized sensitivities of gene activity computed with respect to the resting potential pattern bands has a uniquely complex profile for the discriminator gene- gene 6 (A) where the ordering of the bands in terms of their signed sensitivities goes through a slightly more complex sequence (bottom right inset) compared to the average gene (B). From the perspective of minimal cognition, one could conceive this as a form of shifting of “attention”, where attention is constantly shifted from one band to the other so that the corresponding information could be captured and then integrated before settling on a decision. Due to the symmetry of the model, only the cells in the 2x3 top left quadrant of the full 4x6 tissue were considered for these calculations. Related to Fig. 9.
	Figure S11. Differential effect of gene knockouts on the performance of the model for each of the three input bioelectric patterns. It can be noticed that gene 6 effects the largest drop in performance for the endogenous pattern while contributing very little to processing the other patterns. While gene 4 behaves in a similar fashion as gene 6, genes 2 and 5 behave conversely where they most affect the depolarized and the hyperpolarized pattens while contributing very little to the endogenous pattern. Gene 3 appears to be the most redundant gene of the lot by affecting very little the performance with respect to all three patterns. Related to Fig. 9.
	Figure S12. The timeseries of the discriminator gene following the severing of the connections running from the genes to the Vmem in all cells. It can be seen that the detection of the depolarized voltage pattern is affected, though the behavior with respect to the other patterns are preserved. Related to Fig. 7.
	Figure S14. A more nuanced analysis of the gene activities for the half-and-half and the sharp-endogenous patterns. Panels display gene expression timeseries averaged over the cells in the first 3 columns, the middle 12 columns and the last 3 columns. (A) Response to a voltage pattern with a less skewed ratio of polarization: a half-and-half (step function) voltage pattern where the left half is hyperpolarized, and the right half is depolarized (inset) should result in normal brain development as indicated by the mean gene expression of the discriminator gene, averaged over all cells, discriminating it from the other patterns. (B) Response to a voltage pattern with a more skewed ratio of polarization: a sharpened voltage pattern where the overall pattern is maintained but the hyperpolarized neural plate (central band) is only a third of its original normal size (bottom panel: only the middle two columns are depolarized). Related to Fig. 10.
	Figure 1. Schematic summary of the endogenous voltage prepattern’s control of embryonic brain patterning in Xenopus and an overview of the neural plate circuit model of this process(A) Summary of our previous studies80,87,88,97 showing that the difference (dashed black line) in voltage across the hyperpolarized neural plate and relatively depolarized surrounding ectoderm is essential for driving correct gene expression pattern, large-scale brain morphology, and normal learning behavior.(B) Endogenous embryonic voltage prepattern resulting in normal brain morphology.80,87,88,97(C) Embryonic manipulation of ion fluxes, exposure to teratogens (nicotine and ethanol), or genetic/biochemical disruption (Notch disruption) leading to depolarization of the neural plate erases the critical voltage difference between neural plate and ectoderm, resulting in mispatterned gene expression, and defects in large-scale brain morphology and learning behavior.80,87,88,97,105,106(D) Embryonic manipulation of ion fluxes leading to hyperpolarization of surrounding ectoderm also erases the critical voltage difference between neural plate and ectoderm, resulting in mispatterned gene expression and abnormal large-scale brain morphology.80,97(E) Both local (neural plate) and distant (ectodermal) interventions (manipulating ion fluxes by channel misexpression or drugs targeting ion channels) that restore this critical voltage difference between neural plate and ectoderm, restore gene expression, large-scale brain morphology, and learning behavior.80,87,88,97,105,106(F) A schematic of our model. The overt structure of the model is a 2-dimensional lattice approximately representing the neural plate tissue (shown in the illustration). The circles indicate the individual cells of the tissue, with the colors indicating different voltages corresponding to the endogenous voltage prepattern (blue means depolarized and red, hyperpolarized). Each cell possesses a generic hyperpolarizing channel and depolarizing channel (not shown) and a generic gene regulatory network (gray networks enclosed in squares). The cells exchange gene products and proteins with neighboring cells through the IN-generic intercellular connections (thick gray lines with bidirectional arrowheads) representing paracrine signaling pathways. These collectively represent a plethora of paracrine signaling pathways such as calcium, cAMP, serotonin, integrin, Notch, etc. In this current first iteration model, we generalize them all under paracrine signals as the model is consistent with them all and does not depend on their precise nature. Of note, the IN does not directly allow the exchange of ions but it does so only indirectly via the voltage→gene→gene→voltage pathway (Figure 2A). The voltage and the gene network of a cell influence each other via dedicated connections (thin black lines with unidirectional arrowheads): the voltage acts as an external input to a subset of the genes, affecting their expression and, in turn, the expressions of a subset of the genes can dynamically alter the conductance of the ion channels. The cells in the model are symmetrical, that is, they do not possess a preferred polarity such as that imposed by planar cell polarity. The 4x6 dimension of the lattice shown here is the size that was used during training.(G) Schematic of the input-output map of the pattern discrimination problem; the three input patterns on the left correspond to the embryo schematics displayed in C, B, and D, respectively, with the depolarized (red) and hyperpolarized (blue) voltages corresponding to about −5.2 mV (−10 mV) and −52 mV (−55 mV) in the model (empirical observations80).
	Figure 2. Overview of the mathematical details of the neural plate circuit model and the process of identifying its parameters (training)(A) The equations defining the model, with the variables and parameters mapped to a simplified two-cell version (the rest of the model is symmetrical to this version). The parameters (red) are learned using machine-learning techniques. The bioelectric constants, namely, C, vth, v0, and vT, are described in ref.88 and in Tables S1–S8.(B) The model is trained using a combination of genetic algorithm and gradient descent. The final model referred to throughout this paper is a product of a training loop that starts with a population of randomly parametrized models, simulates all of them, and then modifies them using genetic algorithm and gradient descent to improve their ability to solve the pattern discrimination problem.
	Figure 3. Architecture of the chosen high-performing modelDepicted here is the connectivity of a two-cell version of the full model; the rest of the network is symmetrical to this version (full model parameters described in Tables S1–S8). For clarity, we show only about 20% of the strongest connections (10% of the most positive in purple and 10% of the most negative in orange/brown); edge weights are linearly normalized to the range (−1, 1). The most prominent connections run from the voltage to gene 5 (purple), with a positive weight, and among genes 3 and 5 both within a cell and between cells (brown), with a negative weight. Connections running from genes to voltage do exist in the full model, but they were filtered out due to their weak weights.
	Figure 4. The model not only discriminates between the correct and the incorrect voltage patterns by mapping them to distinct gene expression values but also recapitulates empirically observed development of the voltage pattern(A) Timeseries of the mean normalized gene expression, averaged over all cells and genes, for the three different input conditions depicted on the right: “Depolarized” (all cells are initially depolarized), “Endogenous” (the left and right two columns are initially depolarized with the middle two columns hyperpolarized), or “Hyperpolarized” (all cells are initially hyperpolarized). Insets: (left) discrimination scores of the six individual genes. The red dashed horizontal line indicates the discrimination score expected from a randomly parametrized model. (Right) Gene expression timeseries for gene 6, the gene with the highest discrimination score; the scales of the axes match the scales of the main plot. More detailed timeseries resolved at the level of cells and genes are presented in Figure S2.(B) Top: comparison of the modeled and empirical spatial voltage profiles of a horizontal cross-section of the embryo (schematic) at a point in time when the voltages corresponding to the three different input conditions (treatments) have fully developed (∼200th timestep in the model). Bottom: the spatiotemporal voltage profile of the model depicting the timeseries of Vmem for each cell under each of the three input conditions. Each square in the 4x6 layout of the tissue refers to an individual cell.
	Figure 5. The best fit model flexibly scales to larger tissues when solving the voltage pattern discrimination problemThe timeseries shows the mean expression, averaged over all cells, of the discriminator gene in a scaled up 10x18 model, initialized with the endogenous voltage pattern (inset; left and right six columns are depolarized, and the middle six columns are hyperpolarized), showing that it successfully discriminates between the correct and incorrect voltage patterns. More detailed timeseries resolved at the cell of the level and for larger tissues are presented in Figure S3.
	Figure 6. Differential change in model performance due to knocking out individual or patches of cells in the tissueThe darker the shade of a block, the higher the drop in performance due to knocking out that block. A cell or a patch was “knocked out” by freezing the states of its corresponding genes at zero and dropping their biases to negative infinity, thereby rendering them ineffectual both as inputs and outputs.(A) A 4x6 tissue where individual cells were knocked out.(B) A 10x18 tissue where 2-cell x 3-cell patches (each block is a patch) were knocked out.
	Figure 7. Analytical reconstruction of the gene expression timeseries using only the second-order influence of the voltage pattern on gene expressionA comparison between the normalized original expression timeseries of the discriminator gene (dashed) for the endogenous input pattern and the corresponding reconstruction (solid) obtained from the second-order (Hessian) influence of the endogenous voltage pattern reveals their similarity. The timeseries was reconstructed using the equation g6(t+1)ˆ=g6(t)ˆ+∑j∈TL∑l=1nc∑k=1nccH[6]∂2g6,j(t)∂Gpl∂Gpk(0), where TL refers to the set of cells in the top left quadrant of the tissue and cH was optimized to fit the observed normalized {g6(t)ˆ} with the initial condition g6(0)ˆ=0. Only the cells in the 2x3 top left quadrant of the full 4x6 tissue (dashed boxes) were considered because the model is bilaterally symmetrical both about the vertical and horizontal axes.
	Figure 8. Spatiotemporal dynamics of the second-order control of gene expression by the endogenous voltage patternEach undirected edge represents a pair of cells whose voltages collectively influence the expression of an average discriminator gene in the top left quadrant. The weight of these edges indicates the degree of influence (second-order derivative) exerted by the voltages of the corresponding pairs of cells on the discriminator gene, whereas their colors indicate whether the influence is positive or negative (strong purple means strongly positive and strong orange/brown means strongly negative). These edges represent the summation term in the equation g6(t+1)ˆ=g6(t)ˆ+∑j∈TL∑l=1nc∑k=1nccH[6]∂2g6,j(t)∂Gpl∂Gpk(0), where TL refers to the set of cells in the top left quadrant of the tissue and cH was optimized to fit the observed normalized {g6(t)ˆ} with the initial condition g6(0)ˆ=0. Only the cells in the 2x3 top left quadrant of the full 4x6 tissue (dashed boxes) were considered because the model is bilaterally symmetrical both about the vertical and horizontal axes. For clarity, only about 10% of the observed edges are shown. The arrows running from one band to another serve as a visual illustration of the average weighted and signed influence exerted by the three bands on the target top left quadrant of the pattern.
	Figure 9. A division of labor exists among the genes in terms of their sensitivities to the different spatial scales of the voltage patternThe difference between the first-order (Jacobian) directional derivatives of the gene activities computed with respect to the voltage at the tissue level and the single-cell level tends to be positive for genes i= 2 and 5, negative for genes 3 and 4, and mostly balanced but interspersed with positive spurts for genes 1 and 6. This quantity, known as the net scale sensitivity of a gene, is defined as S(i)=∑j∈TL∑l=1nc1nc∂gi,j(t)∂Gpl(0)−∑j∈TL∂gi,j(t)∂Gpj(0). Here, the first term is a directional derivative and computes the net sensitivity to the tissue, and the second term refers to the single cell. The term 1/nc is a normalization term for computing the directional derivative along the direction of the unit vector, thus enabling the comparison between the two scales of the tissue and the single cell. The corresponding inclinations of sensitivity are toward the tissue level, single-cell level, and an intermediate level, respectively. Due to the symmetry of the model, only the genes contained by the cells in the 2x3 top left quadrant of the full 4x6 tissue were considered for these calculations.
	Figure 10. Model predictions about the responses to two types of non-endogenous input voltage patterns with different polarization ratios simulated in a 10x18 tissueAverage activity levels of the discriminator gene in response to (A) a voltage pattern with a less skewed ratio of polarization in comparison to the endogenous pattern: a half-and-half (step function) voltage pattern where the left half is hyperpolarized and the right half is depolarized (inset); (B) a voltage pattern with a more skewed ratio of polarization in comparison to the endogenous pattern: a sharpened voltage pattern where the overall pattern is maintained but the hyperpolarized neural plate (central band) is only a third of its original normal size (inset).
	Figure 11. Perturbation of voltage in one-half of an embryo, creating a step function voltage pattern, does not cause defects in brain morphology(A) Illustration of Xenopus embryo at stage 3 (four-cell), indicating two dorsal blastomeres as main precursors of eye and brain and two ventral blastomeres as main precursors of non-neural tissues.137,171(B) Normal voltage pattern in control (uninjected) stage ∼15 Xenopus embryos. Top: illustration of expected membrane voltage distribution in stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of CC2-DMPE:DiBAC voltage-reporter-dyes-stained embryos, showing characteristic hyperpolarization in the neural plate (yellow arrows) and surrounding depolarized ectoderm80,87,88 (N = 6) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD.(C) Step function voltage pattern at stage ∼15 in embryos microinjected with Kv1.5 + β-galactosidase mRNA in one dorsal blastomere at four-cell stage. Top: illustration of expected membrane voltage distribution in injected stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of CC2-DMPE: DiBAC voltage-reporter-dyes-stained embryos, showing characteristic hyperpolarization in neural plate (yellow arrow) but hyperpolarized ectoderm only on the injected side (red arrow) (N = 6) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD.(D–G) Representative images of stage 45 tadpoles. (D, E) Tadpoles from uninjected (D) or injected (E) embryos. Blue arrowheads indicate intact nostrils, orange brackets indicate intact forebrain (FB), yellow brackets indicate intact midbrain (MB), cyan brackets indicate intact hindbrain (HB) and intact eyes (e) (F, G) β-Galactosidase expression assessed using X-Gal (blue) in bleached tadpoles. In injected tadpoles (G), X-gal staining is evident in the eye and brain on the injected side (blue arrowheads) but not the uninjected side (magenta arrowheads), validating our targeting of mRNA microinjection. There was no X-gal staining (magenta arrowheads) in uninjected tadpoles (F) (N > 10 tadpoles per experimental group).(H) Quantification of brain morphology defects in stage 45 tadpoles under different injection conditions demonstrates no significant differences among any of the injection conditions. All injections were made into the right dorsal blastomere at 4-cell stage as indicated in the schematic under each graph. Percentage of tadpoles with brain defects for each experimental group are as follows: Controls: 8%, β-galactosidase: 7%, Kv1.5 + β-galactosidase: 8%, and Kir4.1: 6%. Data are mean ± SD, n.s. = non-significant (one-way ANOVA with Tukey’s post-hoc test for n = 3 independent experiments with N > 50 embryos per treatment group per experiment).
	Figure 12. Perturbation of voltage to decrease neural plate hyperpolarized cells while minimally changing the overall voltage pattern leads to significant brain morphology defects(A) Illustration of Xenopus embryo at stage 3 (four-cell) indicating two dorsal blastomeres as main precursors of eye and brain and two ventral blastomeres as main precursors of non-neural tissues.137,171(B) Endogenous voltage pattern in control (uninjected) stage ∼15 embryos. Top: illustration of expected membrane voltage distribution in stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of embryo stained with CC2-DMPE:DiBAC4 voltage reporter dyes, showing characteristic hyperpolarization in the neural plate (yellow arrows) and surrounding depolarized ectoderm80,87,88 (N = 6) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD.(C) Altered voltage pattern in stage ∼15 embryos microinjected with DN-KATP+ β-galactosidase mRNA in one dorsal blastomere at four-cell stage. Top: illustration of expected membrane voltage distribution in injected stage 15 embryo and expected voltage pattern along the dashed line; bottom: representative image of injected embryo stained with CC2-DMPE: DiBAC4 membrane voltage reporter dyes showing reduced region of hyperpolarization in neural plate and narrower hyperpolarization region (empty yellow arrow) (N > 6 embryos per experimental group) and quantification of CC2-DMPE:DiBAC4 images along the dashed line indicated in the illustration along with electrophysiology-based membrane voltage approximations. Data are mean+/− SD.(D–G) Representative images of stage 45 tadpoles. (D, E) Tadpoles from uninjected (D) or microinjected with DN-KATP+ β-galactosidase mRNA (E) embryos. Blue arrowheads indicate intact nostrils; orange, yellow, and cyan brackets indicate intact forebrain (FB), midbrain (MB), and hindbrain (HB), respectively. (e) indicates intact eyes, and orange arrowheads indicate mispatterned brain and eye. (F, G) β-Galactosidase expression assessed using X-Gal (blue) in bleached tadpoles either left uninjected (controls) or co-injected with DN-KATP+ β-galactosidase mRNA. There was no X-gal staining (magenta arrowheads) in uninjected tadpoles (F) In injected tadpoles (G), β-galactosidase was observed in the eye and brain on the injected side (blue arrowheads) but not the uninjected side (magenta arrowheads), confirming expected targeting of microinjected mRNAs. (N > 10 tadpoles per experimental group).(H) DN-KATP + β-galactosidase injection significantly increased the percentage of stage 45 tadpoles with brain morphology defects: uninjected or β-galactosidase controls—both 9%; DN-KATP + β-galactosidase—53%. All injections were made into the right dorsal blastomere at four-cell stage as indicated in the schematic under each graph. Data are mean ± SD, ∗∗p < 0.01, n.s. = non-significant (one-way ANOVA with Tukey’s post-hoc test for n = 3 independent experiments with N > 50 embryos per treatment group per experiment).
	Figure 13. An overview of the current knowledge and gaps in our understanding of Xenopus brain development(A) A schematic of the general state of our knowledge in embryonic brain patterning and the area of focus of this study.(B) A schematic of the functional relationship in which a spatial pattern (the characteristic bell curve voltage distribution, left panel) is transduced into specific gene expression decisions (middle panel, taken from with permission from ref.84) that is required to implement mature brain anatomy (right panel).

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