The transcription factor LHX2 plays a critical role in multiple aspects of neuronal and glial development. In this study, we investigated the long-term electrophysiological consequences of Lhx2 loss in astrocytes in the CA1 region of mouse hippocampus using a genetic strategy that ensured Lhx2 was not disrupted in neurons. These mutant astrocytes exhibited a hyperpolarizing shift in their resting membrane potential, accompanied by a significant increase in input resistance and a significant decrease in input capacitance, together indicating altered biophysical properties compared with astrocytes in control mice. However, we found no significant alterations in the intrinsic electrophysiological characteristics of hippocampal CA1 neurons in mice with Lhx2 mutant astrocytes compared with controls. Collectively, we identify LHX2 as a regulator of intrinsic membrane properties of hippocampal astrocytes.NEW & NOTEWORTHY Astrocytes play a vital role in regulating neural circuit homeostasis, yet the transcriptional mechanisms governing their electrophysiological properties remain underexplored. Here, we identify the LIM-homeodomain transcription factor LHX2 as a key regulator of intrinsic electrical properties of astrocytes in the hippocampus. We demonstrate that deletion of Lhx2 from postnatal astrocytes and their progenitors alters astrocytic membrane biophysics in adulthood, but this did not have noncell-autonomous effects on the intrinsic properties of CA1 pyramidal neurons.

The dentate gyrus (DG) of the hippocampus exhibits striking anatomical and functional heterogeneity along its dorsoventral axis, yet the intrinsic electrophysiological diversity of its principal excitatory neurons, the granule cells, across the dorsoventral axis remains unexplored. Here, we systematically examined the electrophysiological properties of DG granule cells across the dorsal, intermediate, and ventral regions of the rat hippocampus. We found a progressive increase in input resistance, impedance amplitude, and firing rate of granule cells, accompanied by a gradual slowdown in repolarization kinetics of their action potentials along the dorsal-to-ventral axis. Our analyses demonstrated that granule cells acted as class I integrators that lacked strong resonance properties across the dorsoventral axis. We performed pairwise correlation and dimensionality reduction analyses to reveal weak dependencies across physiological measurements and the absence of distinct clusters for dorsal, intermediate, or ventral granule cells. Importantly, blade-specific analyses of granule cell physiology revealed that all measurements manifested pronounced heterogeneities even within a given dorsoventral section and a specific blade. Strikingly, ventral granule cells in the infrapyramidal blade manifested higher firing rates compared to their counterparts in the suprapyramidal blade. These blade-specific differences were limited to the ventral granule cells, with negligible distinctions between granule cells in the two blades of either dorsal or intermediate hippocampus. Together, our findings unveil a progressive increase in excitability of DG granule cells along the dorsal-to-ventral axis and a blade-specific granularity of firing properties, adding new dimensions to the several known anatomical, molecular, and behavioral differences across the hippocampal dorsoventral axis.

Local field potentials (LFPs) are compound signals that represent the dynamic flow of information across the brain, which have been historically associated with chemical synaptic inputs. How do gap junctional inputs onto active compartments shape LFPs? We developed a methodology to record extracellular potentials associated with different patterns of gap junctional inputs onto conductance-based models. We found that synchronous inputs through chemical synapses yielded a negative deflection in proximal extracellular electrodes whereas those onto gap junctions manifested a positive deflection. Importantly, we observed extracellular dipoles only when inputs arrived through chemical synapses but not with gap junctions. Remarkably, hyperpolarization-activation cyclic nucleotide-gated channels, which typically conduct inward currents, mediated outward currents triggered by the fast voltage transition caused by synchronous inputs. With rhythmic inputs at different frequencies arriving through gap junctions, we found strong suppression of LFP power at higher frequencies as well as frequency-dependent differences in the spike phase associated with the LFP when compared to respective chemical synaptic counterparts. All observed differences in LFP were mediated by the relative dominance of synaptic currents vs . voltage-driven transmembrane currents with chemical synapses vs . gap junctions, respectively. Our analyses unveil a hitherto unknown role for active dendritic gap junctions in shaping extracellular potentials.

The continuous attractor network (CAN) model explains grid-patterned firing and path integration in the entorhinal cortex, yet the impact of sensory noise in velocity inputs remains unexplored. In addressing this, we introduced varying levels of noise to the velocity inputs impinging on a 2D CAN model driven a virtual animal traversing an arena. We estimated animal position from network activity and quantified position accuracy as the difference between real and estimated positions. We performed all simulations using several trajectories, as grid scores and position accuracy showed pronounced trajectory-to-trajectory variability even without noise. We found that low levels of sensory noise were beneficial to grid-field formation, particularly for trajectories that failed to generate grid patterns in noise-free conditions. For trajectories exhibiting grid-patterned activity without noise, low levels of noise improved position estimation accuracy. In contrast, high levels of sensory noise impaired position estimates and grid-patterned activity. These results demonstrate stochastic resonance in a 2D CAN model, where an optimal level of sensory noise enabled grid-patterned activity and enhanced position accuracy. Motivated by the proposed error-correcting role of border cells, we introduced north and east border cells that were connected to grid cells based on co-activity patterns. Interestingly, while border inputs enabled grid field formation in cases where grid patterns were previously absent, their effect on position accuracy was marginal. Together, our analyses suggest that biological CANs could evolve to yield optimal performance in the presence of noise, which could serve as a stabilizing factor yielding functional robustness through stochastic resonance.

ABSTRACT A key challenge in understanding spatial navigation and memory is explaining how hippocampal networks sustain robust spatial information transfer despite pronounced trial-to-trial variability and pervasive neural-circuit heterogeneities. Although hippocampal heterogeneities and physiological variability are well-characterized, circuit-scale understanding of stable information transfer in recurrent place-cell networks remains limited. Here, we first show that even recurrent networks composed of intrinsically identical neurons and receiving identical place-field inputs express pronounced neuron-to-neuron variability in spatial tuning profiles, place-field widths, subthreshold ramp amplitudes, and spatial information transfer. Introduction of intrinsic within-type heterogeneities to excitatory and inhibitory neurons further increased diversity in firing properties, but strikingly improved robustness of spatial information transfer under high trial-to-trial variability. Although strengthening inhibition expectedly narrowed place fields and reduced firing across all networks, the impact of inhibition on information-transfer profiles was stronger in heterogeneous networks manifesting high degree of trial-to-trial variability. Across networks, increasing trial-to-trial variability reduced information transfer and shifted the spatial location of peak information from the high-slope regions of the tuning curve to its peak-firing location. Finally, incorporating afferent heterogeneities allowed neurons to be tuned to distinct place-field centers, reducing peak information values while amplifying neuron-to-neuron diversity in information-transfer profiles. Together, we demonstrate that excitation-inhibition balance, trial-to-trial variability, within-type heterogeneities, and afferent input diversity jointly regulate spatial information transfer in recurrent place-cell circuits. We also highlight intrinsic heterogeneities as substrates for enhanced robustness of spatial coding against perturbations. Importantly, the convergence of multiple disparate mechanisms in yielding similar information-transfer profiles underscores degeneracy as a fundamental organizing principle in neural-circuit physiology.

ABSTRACT Motivation Information flow and temporal coding in cortical circuits depend critically on the reliable transmission of precisely timed synchronous spike patterns. Although cortical assemblies achieve such transmission despite pronounced intrinsic heterogeneities and stochastic high-conductance states, the mechanisms underlying effective synchrony propagation under in vivo conditions remain poorly understood. Methodology In this study, we address this gap using large-scale, conductance-based models of excitatory and inhibitory neurons organized into feedforward synfire chains operating in noisy, high-conductance regimes. Using independent stochastic search algorithms, we first identified physiologically valid heterogeneous populations of cortical neurons. Both excitatory and inhibitory populations exhibited cellular-scale degeneracy, whereby distinct combinations of biophysically identified molecular components produced signature physiological characteristics. We then constructed synfire chains with varying degrees of heterogeneity using these populations and assessed the propagation of different spike packets across neuronal assemblies. Results We found synchrony propagation to be inherently probabilistic, revealing a stochastic separatrix that separated input patterns that consistently succeeded from those that consistently failed in propagation. The stochastic nature of this separatrix highlighted a critical role for background synaptic fluctuations, defining a regime in which identical inputs alternately propagated or failed across trials solely due to stochastic background activity. Comparing networks with different degrees of intrinsic heterogeneity, we found that increasing heterogeneity did not alter mean propagation efficacy but reduced network-to-network variability, indicating a stabilizing role for intrinsic diversity. Strikingly, when we tested the impact of neuronal intrinsic properties on synchrony propagation, hyperpolarization-activated cyclic nucleotide-gated (HCN) channels emerged as robust enhancers of synchrony propagation across all heterogeneity regimes. Mechanistically, the slow restorative kinetics of HCN conductances narrowed the temporal window for spike initiation, sharpening output synchrony, and improving propagation reliability. This effect was abolished when HCN kinetics were accelerated, underscoring the importance of the slow negative feedback mediated by these channels. Implications Together, our analyses identify HCN channels as key regulators of synchronous information transfer and reveal strong interactions among intrinsic conductances, input characteristics, neuronal heterogeneity, and stochastic background activity in shaping cortical synchrony propagation. The ability of diverse cellular and network configurations to achieve similar propagation efficacy further highlights degeneracy as a fundamental principle governing robust and flexible neural computation.

SUMMARY Artificial recurrent networks are powerful models for studying neural dynamics and representations underlying complex cognitive tasks. However, the impact of neural-circuit heterogeneities on learning, dynamics, robustness, and generalization in these networks remains poorly understood. Here, we systematically investigated the impact of graded intrinsic heterogeneities in artificial recurrent networks trained on different cognitive tasks using reward- modulated Hebbian learning. Across networks trained with distinct hyperparameters and different levels of intrinsic heterogeneity, we observed pronounced network-to-network and task-to-task variability in training convergence, error dynamics during training, and task performance. These effects were strongly task dependent, with memory-dependent tasks exhibiting greater sensitivity to heterogeneity than memoryless tasks. We assessed these networks for robustness to multiple forms of graded post-training perturbations. Perturbations to intrinsic time constant distributions altered network dynamics, but had limited impact on final task accuracy in most cases. In contrast, perturbations to initial conditions, exploratory activity impulses, or task epoch durations strongly affected memory-dependent tasks. Among all perturbations, synaptic jitter was consistently the most detrimental, impairing performance across all tasks and heterogeneity levels. Importantly, despite such pronounced impact of heterogeneities, none of the metrics (spanning training, performance, dynamics, and robustness) varied monotonically with the level of training heterogeneity, instead showing additional dependencies on task demands, network configuration, and perturbation type. Finally, networks trained on a single task were able to perform structurally related untrained tasks, but failed on fundamentally distinct tasks. Strikingly, similar task performances emerged from divergent activity trajectories across networks and training conditions, together revealing pronounced functional degeneracy in network dynamics. Collectively, our findings establish that heterogeneous recurrent networks operate in a complex systems regime, where robust function emerges from non-unique, task-specific interactions among hyperparameters, dynamics, and heterogeneities. Our analyses emphasize the need for population- of-networks approaches that focus on interactions among multiple forms of neural heterogeneities in shaping learning and computation.

In the developing cerebral cortex, astrocytes arise from progenitors in the ventricular and subventricular zones (V-SVZ), and also from local proliferation within the parenchyma. In the mouse neocortex, astrocytes that occupy upper layers (UL) versus deep layers (DL) are known to be distinct populations in terms of molecular and morphological features. The transcription factor LHX2 is expressed both in V-SVZ gliogenic progenitors and in differentiated astrocytes throughout development and into adulthood. Here, we show that loss of Lhx2 at birth results in an increased astrocyte proliferation in the UL but not the DL of the cortex in the first postnatal week. Consistent with this, transcriptomic signatures of UL astrocytes increase. By 3 months, Lhx2 mutant astrocytes display upregulation of GFAP, and transcriptomic signatures associated with 'reactive' astrocytes, in the absence of injury. These results demonstrate a role for Lhx2 in regulating proliferation and molecular features of cortical astrocytes.

In the mammalian neocortex, the two hemispheres communicate via the corpus callosum. We investigated mechanisms regulating dendritic arbors and spines of callosal neurons. The transcription factor LIM Homeodomain 2 (Lhx2), a key regulator of cortical development, is expressed in postmitotic layer II/III neurons and their progenitors. Loss of Lhx2 in either population caused similar but distinct phenotypes: reduced dendritic arbors, altered spine morphology, and changed electrophysiological properties. Morphometric defects were more severe when Lhx2 was disrupted in progenitors and were recapitulated by its specific loss in basal progenitors. Lhx2 loss in progenitors aberrantly up-regulated Neurog2 in postmitotic neurons, and Neurog2 knockdown partially rescued the phenotype. Loss of Lhx2 at either stage also up-regulated Wnt signaling pathway genes. The mutant phenotype was mimicked by constitutive activation of β-CATENIN in postmitotic neurons. Our findings reveal previously unidentified LHX2-dependent mechanisms of dendritic morphogenesis, highlighting its temporally dynamic and diverse roles in neocortical development.

Background and motivations : The Marr-Albus theory postulates that pattern separation is realized by divergent feedforward excitatory connectivity. Yet, there are several lines of evidence for strong but differential regulation of pattern separation by local circuit connectivity, even when feedforward connectivity is divergent. What are the relative contributions of divergent feedforward connectivity and local circuit interactions to pattern separation? How do we reconcile the contrasting lines of evidence on local circuit regulation of pattern separation in circuits endowed with divergent feedforward connectivity? In this study, we quantitatively address these questions in a population of heterogeneous dentate gyrus (DG) networks, where we enforced feedforward connectivity to be identically divergent. Methodology : We generated tens of thousands of random spiking neuronal models to arrive at thousands of non-repeating heterogeneous single-neuron models of four different DG neuronal subtypes, each satisfying their respective functional characteristics. We connected these heterogeneous populations of neurons with subtype proportions and local connectivity that reflected the DG microcircuit. In a second level of unbiased search, we generated 20,000 identical networks that differed from each other only in their synaptic weight values. Thus, within the Marr-Albus framework, these networks that were identical in terms of neuronal composition and divergent feedforward connectivity should all be capable of performing effective pattern separation. To test this, we fed these networks with morphed sets of input patterns, recorded granule cell outputs, and computed similarity metrics based on correlation measures across input or output patterns. We developed novel quantitative metrics for pattern separation from plots of output similarity vs. input similarity and validated each network with bounds on these metrics. Results : Despite being identical in terms of divergent feedforward connectivity, we found only a very small proportion (47 of 20,000 or 0.23%) of the randomly generated networks to manifest effective pattern separation. We tested the specific contributions of interneurons by assessing pattern separation in all pattern-separating networks after individually deleting each of the three interneuron subtypes. Strikingly, we found pronounced network-to-network variability in how each interneuron subtype contributed to granule cell sparsity and pattern separation. We traced this variability to differences in local synaptic connectivity, which also resulted in network-to-network variability in firing rates and sparsity of different interneurons. Finally, we found heterogeneous DG networks to be more resilient to synaptic jitter compared to their homogeneous counterparts, with specific reference to pattern separation computed through average firing rate correlations. Implications : Our population-of-networks approach clearly shows that divergent connectivity of afferent inputs does not guarantee pattern separation in DG networks. Instead, we demonstrate strong yet variable roles for interneurons and local connectivity in implementing pattern separation. Importantly, our analyses unveil degeneracy in DG circuits, whereby similar pattern separation efficacy was achieved through disparate local-circuit interactions. These observations, alongside network-to-network variability in dependencies on different interneurons, strongly advocate the complex adaptive systems approach as a unifying framework to study DG pattern separation. ### Competing Interest Statement The authors have declared no competing interest.

Background and motivation Local field potentials (LFPs) are compound signals comprising synaptic currents and several transmembrane currents from active structures, which represent the dynamic flow of information across the brain. Although LFP analyses have remained largely limited to chemical synaptic inputs, neurons and other cell types also receive gap junctional inputs that play essential roles in neuronal and network physiology. Gap junctional inputs have been historically excluded from LFP analyses because, unlike synaptic receptors, these inputs are not mediated by transmembrane currents that involve the extracellular space. However, the voltage response to gap junctional inputs onto active compartments triggers several transmembrane currents across the neuron. Therefore, two fundamental questions required for enhanced accuracy of LFP interpretations are: (i) Do gap junctional inputs onto active compartments contribute to LFPs? (ii) Are there differences in extracellular signatures associated with gap junctional vs . chemical synaptic inputs onto active compartments? Methodology We built morphologically realistic conductance-based neuronal models and placed a 3D array of extracellular electrodes spanning the somato-dendritic stretch. We employed different types of inputs: (i) synchronous; (ii) random; and (iii) rhythmic (1–128 Hz). We computed LFPs at all electrodes and analyzed the spatiotemporal profiles of intra- and extra-cellular voltages for several model configurations, involving different input types, with active vs . passive dendrites, with gap junctions vs . chemical synapses, and in the presence vs . absence of different ion channels. Results We demonstrate a striking reversal in the polarity of extracellular potentials associated with synchronous inputs through chemical synapses vs . gap junctions onto active dendrites. Whereas synchronous inputs through chemical synapses yielded a negative deflection in proximal electrodes, those onto gap junctions manifested a positive deflection. Importantly, we observed extracellular dipoles only when inputs arrived through chemical synapses, but not with gap junctions. Remarkably, the slow hyperpolarization-activation cyclic nucleotide-gated (HCN) channels, which typically conduct inward currents, mediated outward currents triggered by the fast voltage transition caused by synchronous inputs. With random inputs, extracellular potentials in proximal electrodes were largely negative with chemical synapses but were biphasic with gap junctional inputs. Finally, with rhythmic inputs arriving through gap junctions, we found strong suppression of LFP power at higher frequencies. There were frequency-dependent differences in the spike phase associated with the LFP, depending on whether inputs arrived through gap junctions or chemical synapses. LFP differences across all input types were mediated by the relative dominance of synaptic currents vs . voltage-driven transmembrane currents with chemical synapses vs . gap junctions, respectively. Implications Our analyses unveil a prominent role for gap junctional connections in shaping the spatiotemporal and spectral profiles of extracellular potentials, with critical implications for polarities and spatial spread of LFPs. The stark dichotomies in extracellular signatures associated with gap junctional vs . chemical synaptic inputs imply that conclusions could be erroneous if cells were incorrectly assumed to be exclusively receiving chemical synaptic inputs. ### Competing Interest Statement The authors have declared no competing interest.

Much effort has been spent clustering neurons into transcriptomic or functional cell types and characterizing the differences between them. Beyond subdividing neurons into categories, we must recognize that no two neurons are identical and that graded physiological or transcriptomic properties exist within cells of a given type. This often overlooked “within-type” heterogeneity is a specific neuronal implementation of what statistical physics refers to as “disorder” and exhibits rich computational properties, the identification of which may shed crucial insights into theories of brain function. In this perspective article, we address this gap by highlighting theoretical frameworks for the study of neural tissue heterogeneity and discussing the benefits and implications of within-type heterogeneity for neural network dynamics, computation, and self-organization.

Pattern separation, the ability of a network to distinguish similar inputs by transforming them into distinct outputs, was postulated by the Marr-Albus theory to be realized by divergent feedforward excitatory connectivity. Yet, there is evidence for strong but differential regulation of pattern separation by local circuit connectivity. How do we reconcile the conflicting views on local-circuit regulation of pattern separation in circuits receiving divergent feedforward connectivity? Here, we quantitatively examined a population of heterogeneous dentate gyrus (DG) spiking networks where identically divergent feedforward connectivity was enforced. We generated 20 000 random DG networks constructed with thousands of functionally validated, heterogeneous single-neuron models of 4 different DG neuronal subtypes. We recorded network outputs to morphed sets of input patterns and applied quantitative metrics that we developed to assess pattern separation performance of each network. Surprisingly, only 47 of these 20 000 networks (0.23%) manifested effective pattern separation showing that divergent feedforward connectivity alone does not guarantee pattern separation. Instead, our analyses unveiled strong contributions from the 3 interneuron subtypes toward granule cell sparsity and pattern separation, with pronounced network-to-network variability in such contributions. We traced this variability to differences in local synaptic weights across pattern-separating networks, highlighting synaptic degeneracy as a key mechanism that explains diversity in interneuronal regulation of pattern separation. Finally, we found heterogeneous DG networks to be more resilient to synaptic jitter compared to their homogeneous counterparts. Together, our findings reconcile conflicting evidence by revealing degeneracy in DG circuits, whereby similar pattern separation efficacy can arise through diverse interactions among granule cells and interneurons.

Concepts from network science and graph theory, including the framework of network motifs, have been frequently applied in studying neuronal networks and other biological complex systems. Network-based approaches can also be used to study the functions of individual neurons, where cellular elements such as ion channels and membrane voltage are conceptualized as nodes within a network, and their interactions are denoted by edges. Network motifs in this context provide functional building blocks that help to illuminate the principles of cellular neurophysiology. In this review we build a case that network motifs operating within neurons provide tools for defining the functional architecture of single-neuron physiology and neuronal adaptations. We highlight the presence of such computational motifs in the cellular mechanisms underlying action potential generation, neuronal oscillations, dendritic integration, and neuronal plasticity. Future work applying the network motifs perspective may help to decipher the functional complexities of neurons and their adaptation during health and disease.

Background The continuous attractor network (CAN) model has been effective in explaining grid-patterned firing in the rodent medial entorhinal cortex, with strong lines of experimental evidence and widespread utilities in understanding spatial navigation and path integration. A surprising lacuna in CAN analyses is the paucity of quantitative studies on the impact of afferent sensory noise on path integration. Here, we evaluate the impact of afferent sensory noise on grid-patterned firing and on the accuracy of position estimates derived from network pattern flow velocity. Motivated by the ability of border cells to act as an error-correction mechanism, we also assess the impact of interaction between afferent noise and border cell inputs on CAN performance. Methodology We used an established 2D CAN model that received velocity inputs from a virtual animal traversing a 2D arena to generate grid-patterned firing. We estimated network pattern flow velocity from network activity and used that to compute an activity-based position estimate at each time step. We tracked the difference between the real and the estimated positions as a function of time and called it the deviation in integrated path (DIP). We defined afferent sensory noise to be additive Gaussian, with different noise levels achieved by changing the variance. We introduced north and east border cells and connected them to grid cells based on co-activity patterns. For different levels of noise, we computed DIP and metrics for grid-patterned activity in the presence vs . absence of border cells. Importantly, to avoid potential bias owing to the use of a single trajectory in computing these measurements, we performed all simulations across 50 different trajectories. Results The computed grid scores and position accuracy (as DIP) showed pronounced trajectory-to-trajectory variability, even in a noise-free network. With the introduction of sensory noise, the variability prevailed and unveiled a dichotomous impact of afferent sensory noise on position accuracy vs . grid-patterned activity. Specifically, low levels of sensory noise improved position estimation accuracy without altering the ability of the network to generate grid-patterned activity. In contrast, high levels of sensory noise impaired position estimates as well as grid-patterned activity, although position estimates were more sensitive to sensory noise compared to grid-patterned activity. The stochastic resonance observed in the relationship between position accuracy and sensory noise level was partially explained by the interaction of noisy inputs with the rectification nonlinearity in the neural transfer function. Finally, across noise levels, pronounced trajectory-to-trajectory variability in grid-score and position accuracy was observed with the addition of border inputs. Across the population of trajectories, addition of border inputs yielded modest changes in both measurements across noise levels. Implications Our analyses demonstrate that the robustness of grid-patterned activity in CAN models to noise does not extend to other functions of the CAN model. Stochastic resonance with reference to position estimation and sensory noise implies that biological CANs could evolve to yield optimal performance (path integration) in the presence of noise in biological sensory systems. An important methodological implication that emerges from our observations is the critical need to account for trajectory-to-trajectory variability in position estimates and path integration. Given the pronounced nature of trajectory-to-trajectory variability, conclusions based on a single trajectory are bound to be erroneous thereby warranting analyses with multiple trajectories. Together, our analyses unveil important roles for sensory noise in improving position estimates obtained from activity in CAN models. ### Competing Interest Statement The authors have declared no competing interest.

Degeneracy is defined as multiple sets of solutions that can produce very similar system performance. Degeneracy is seen across phylogenetic scales, in all kinds of organisms. In neuroscience, degeneracy can be seen in the constellation of biophysical properties that produce a neuron's characteristic intrinsic properties and/or the constellation of mechanisms that determine circuit outputs or behavior. Here, we present examples of degeneracy at multiple levels of organization, from single-cell behavior, small circuits, large circuits, and, in cognition, drawing conclusions from work ranging from bacteria to human cognition. Degeneracy allows the individual-to-individual variability within a population that creates potential for evolution.

Complex systems are neither fully determined nor completely random. Biological complex systems, including single neurons, manifest intermediate regimes of randomness that recruit integration of specific combinations of functionally segregated subsystems. Such emergence of biological function provides the substrate for the expression of degeneracy, the ability of disparate combinations of subsystems to yield similar function. Here, we present evidence for the expression of degeneracy in morphologically realistic models of dentate gyrus granule cells (GC) through functional integration of disparate ion-channel combinations. We performed a 45-parameter randomized search spanning 16 active and passive ion channels, each biophysically constrained by their gating kinetics and localization profiles, to search for valid GC models. Valid models were those that satisfied 17 sub- and supra-threshold cellular-scale electrophysiological measurements from rat GCs. A vast majority (>99%) of the 15,000 random models were not electrophysiologically valid, demonstrating that arbitrarily random ion-channel combinations wouldn't yield GC functions. The 141 valid models (0.94% of 15,000) manifested heterogeneities in and cross-dependencies across local and propagating electrophysiological measurements, which matched with their respective biological counterparts. Importantly, these valid models were widespread throughout the parametric space and manifested weak cross-dependencies across different parameters. These observations together showed that GC physiology could neither be obtained by entirely random ion-channel combinations nor is there an entirely determined single parametric combination that satisfied all constraints. The complexity, the heterogeneities in measurement and parametric spaces, and degeneracy associated with GC physiology should be rigorously accounted for, while assessing GCs and their robustness under physiological and pathological conditions.

In the mammalian neocortex, excitatory neurons that send projections via the corpus callosum are critical to integrating information across the two brain hemispheres. The molecular mechanisms governing the development of the dendritic arbours and spines of these callosal neurons are poorly understood, yet these features are critical to their physiological properties. LIM Homeodomain 2 ( Lhx2 ), a regulator of fundamental processes in cortical development, is expressed in postmitotic callosal neurons occupying layer II/III of the neocortex and also in their progenitors in the embryonic day (E) 15.5 ventricular zone of the mouse neocortex. We tested whether this factor is essential for dendritic arbour configuration and spine morphogenesis of layer II/III neurons. Here, we report loss of Lhx2 either in postmitotic callosal neurons or their progenitors, resulting in shrunken dendritic arbours and perturbed spine morphology. In postmitotic neurons, we identified that LHX2 regulates dendritic and spine morphogenesis via the canonical Wnt/β Catenin signalling pathway. Constitutive activation of this pathway in postmitotic neurons recapitulates the Lhx2 loss-of-function phenotype. In E15.5 progenitors, we identified that bHLH transcription factor Neurog2 mediates LHX2 function in regulating dendritic and spine morphogenesis. We show that Neurog2 expression increases upon loss of Lhx2 and that shRNA-mediated Neurog2 knockdown rescues the loss of Lhx2 phenotype. Our study uncovers novel LHX2 functions in cortical circuit assembly consistent with its temporally dynamic and multifunctional roles in development.### Competing Interest StatementThe authors have declared no competing interest.

Background and motivation Brain rhythms have been postulated to play central roles in animal cognition. A prominently reported dichotomy of hippocampal rhythms, driven primarily by historic single- strata recordings, assigns theta-frequency oscillations (4–12 Hz) and ripples (120–250 Hz) to be exclusively associated with preparatory and consummatory behaviors, respectively. However, due to the differential power expression of these two signals across hippocampal strata , reports of such exclusivity require validation through simultaneous multi- strata recordings and cross- strata analysis of these oscillatory patterns. Methodology We assessed co-occurrence of theta-frequency oscillations with ripples in multi-channel recordings of extracellular potentials across hippocampal strata from foraging rats. We detected all ripple events from an identified stratum pyramidale ( SP ) channel based on rigorous thresholds relating to the spectro-temporal and spatial characteristics of ripples. We then defined theta epochs based on theta oscillations detected from each of the different channels spanning the SP to the stratum lacunosum-moleculare ( SLM ) through the stratum radiatum ( SR ). We calculated the proportion of ripples embedded within theta epochs. Results We found ∼20% (across rats) of ripple events (in SP ) to co-occur with theta epochs identified from SR / SLM channels, defined here as theta ripples . All characteristics of theta ripples were comparable with ripples that occurred in the absence of theta oscillations. Furthermore, the power of theta oscillations in the immediate vicinity of theta ripples was similar to theta power across identified theta epochs, together validating the identification process of theta ripples. Strikingly, when theta epochs were instead identified from the SP channel, such co-occurrences were significantly lower in number. The reduction in the number of theta ripples was consequent to progressive reduction in theta power along the SLM-SR-SP axis. We assessed the behavioral state of rats during ripple events and found most theta ripples to occur during immobile periods. We confirmed that across sessions and rats, the theta power observed during exploratory theta epochs was comparable with theta power during immobile theta epochs. In addition, the progressive reduction in theta power along the SLM-SR-SP axis was common to both exploratory and immobile periods. Finally, we found a strong theta-phase preference of theta ripples within the third quadrant [3π/2–2π] of the associated theta oscillation. Implications Our analyses provide direct quantitative evidence for the occurrence of ripple events nested within theta oscillations in the rodent hippocampus. These analyses emphasize that the prevalent dichotomy about the manifestation of theta-frequency oscillations and ripples needs to be reevaluated, after explicitly accounting for the differential stratum -dependent expression of these two oscillatory patterns. The prevalence of theta ripples expands the potential roles of ripple-frequency oscillations to span the continuum of encoding, retrieval, and consolidation, achieved through interactions with theta oscillations.