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Neocortex, a new and rapidly evolving brain structure in mammals, has a similar layered architecture in species over a wide range of brain sizes. Larger brains require longer fibers to communicate between distant cortical areas; the volume of the white matter that contains long axons increases disproportionally faster than the volume of the gray matter that contains cell bodies, dendrites, and axons for local information processing, according to a power law. The theoretical analysis presented here shows how this remarkable anatomical regularity might arise naturally as a consequence of the local uniformity of the cortex and the requirement for compact arrangement of long axonal fibers. The predicted power law with an exponent of 4/3 minus a small correction for the thickness of the cortex accurately accounts for empirical data spanning several orders of magnitude in brain sizes for various mammalian species, including human and nonhuman primates.

Although the orientation of an arm in space or the static view of an object may be represented by a population of neurons in complex ways, how these variables change with movement often follows simple linear rules, reflecting the underlying geometric constraints in the physical world. A theoretical analysis is presented for how such constraints affect the average firing rates of sensory and motor neurons during natural movements with low degrees of freedom, such as a limb movement and rigid object motion. When applied to nonrigid reaching arm movements, the linear theory accounts for cosine directional tuning with linear speed modulation, predicts a curl-free spatial distribution of preferred directions, and also explains why the instantaneous motion of the hand can be recovered from the neural population activity. For three-dimensional motion of a rigid object, the theory predicts that, to a first approximation, the response of a sensory neuron should have a preferred translational direction and a preferred rotation axis in space, both with cosine tuning functions modulated multiplicatively by speed and angular speed, respectively. Some known tuning properties of motion-sensitive neurons follow as special cases. Acceleration tuning and nonlinear speed modulation are considered in an extension of the linear theory. This general approach provides a principled method to derive mechanism-insensitive neuronal properties by exploiting the inherently low dimensionality of natural movements.

Sensory and motor variables are typically represented by a population of broadly tuned neurons. A coarser representation with broader tuning can often improve coding accuracy, but sometimes the accuracy may also improve with sharper tuning. The theoretical analysis here shows that the relationship between tuning width and accuracy depends crucially on the dimension of the encoded variable. A general rule is derived for how the Fisher information scales with the tuning width, regardless of the exact shape of the tuning function, the probability distribution of spikes, and allowing some correlated noise between neurons. These results demonstrate a universal dimensionality effect in neural population coding.

Coarse codes are widely used throughout the brain to encode sensory and motor variables. Methods designed to interpret these codes, such as population vector analysis, are either inefficient (the variance of the estimate is much larger than the smallest possible variance) or biologically implausible, like maximum likelihood. Moreover, these methods attempt to compute a scalar or vector estimate of the encoded variable. Neurons are faced with a similar estimation problem. They must read out the responses of the presynaptic neurons, but, by contrast, they typically encode the variable with a further population code rather than as a scalar. We show how a nonlinear recurrent network can be used to perform estimation in a near-optimal way while keeping the estimate in a coarse code format. This work suggests that lateral connections in the cortex may be involved in cleaning up uncorrelated noise among neurons representing similar variables.

Location-Controlled Oscillator Model This repository contains the IPython Notebook files that were used to develop the location-controlled oscillator (LCO) model that I presented at the Society for Neuroscience 2014 meeting: J. D. Monaco, K. Zhang, and H. T. Blair. (2014) Spatial rate/phase codes provide landmark-based error correction in a temporal model of theta cells. Society for Neuroscience 2014, Washington, D.C. 752.07. Abstract The spatial firing patterns of place cells in hippocampus and grid cells in entorhinal cortex form a spatial representation that is stable during active navigation but also able to encode changes in external landmarks or environmental structure. One class of model that has been investigated as a possible mechanism for generating these spatial patterns relies on temporal synchronization between theta cells, which fire strongly with the septohippocampal theta rhythm (6–10 Hz) and are found throughout the hippocampal formation, that act as velocity-controlled oscillators. However, a critical problem for these models is that the oscillatory interference patterns that they generate become unstable in the presence of phase noise and errors in self-motion signals. Previous studies have proposed hybridizing temporal models with attractor network models or integrating environmental feedback from sensory cues. Preliminary data from subcortical regions in rats suggest that some theta cells exhibit spatially selective firing similar to hippocampal place fields or entorhinal/subicular boundary fields. These cells also demonstrate a consistent phase relationship across space, relative to ongoing hippocampal theta and to other simultaneously recorded cells, that is correlated with the firing rate at a given location. Inspired by this data, we present a novel synchronization model in which place cells or boundary-vector cells provide a stable, landmark-based excitatory input that drives a rate-to-phase mechanism to generate a population of cells that act as location-controlled oscillators. These cells fire preferentially at theta phases that are specific to a given location, determined by the presence of external landmarks. We show that these location-controlled oscillators provide a stable spatial reference that corrects phase errors in velocity-controlled oscillators, thus preventing the encoded position from drifting with respect to the environment and the actual position of the animal. Thus landmark-based rate/phase correlations in extrahippocampal areas may provide the sensory feedback required by temporal models of neural representations of space.

In this paper we review several lines of recent work aimed at developing practical methods for adaptive on-line stimulus generation for sensory neurophysiology. We consider various experimental paradigms where on-line stimulus optimization is utilized, including the classical \textit{optimal stimulus} paradigm where the goal of experiments is to identify a stimulus which maximizes neural responses, the \textit{iso-response} paradigm which finds sets of stimuli giving rise to constant responses, and the \textit{system identification} paradigm where the experimental goal is to estimate and possibly compare sensory processing models. We discuss various theoretical and practical aspects of adaptive firing rate optimization, including optimization with stimulus space constraints, firing rate adaptation, and possible network constraints on the optimal stimulus. We consider the problem of system identification, and show how accurate estimation of nonlinear models can be highly dependent on the stimulus set used to probe the network. We suggest that optimizing stimuli for accurate model estimation may make it possible to successfully identify nonlinear models which are otherwise intractable, and summarize several recent studies of this type. Finally, we present a two-stage stimulus design procedure which combines the dual goals of model estimation and model comparison and may be especially useful for system identification experiments where the appropriate model is unknown beforehand. We propose that fast, on-line stimulus optimization enabled by increasing computer power can make it practical to move sensory neuroscience away from a descriptive paradigm and towards a new paradigm of real-time model estimation and comparison.

Attractor networks are a popular computational construct used to model different brain systems. These networks allow elegant computations that are thought to represent a number of aspects of brain function. Although there is good reason to believe that the brain displays attractor dynamics, it has proven difficult to test experimentally whether any particular attractor architecture resides in any particular brain circuit. We review models and experimental evidence for three systems in the rat brain that are presumed to be components of the rat's navigational and memory system. Head-direction cells have been modeled as a ring attractor, grid cells as a plane attractor, and place cells both as a plane attractor and as a point attractor. Whereas the models have proven to be extremely useful conceptual tools, the experimental evidence in their favor, although intriguing, is still mostly circumstantial.

The anterodorsal nucleus of the thalamus contains a high percentage of head-direction cells whose activities are correlated with an animal's directional heading in the horizontal plane. The firing of head-direction cells could involve self-sustaining reverberating activity in a recurrent network, but the thalamus by itself lacks strong excitatory recurrent synaptic connections to sustain tonic reverberating activity. Here we examined whether a single thalamic neuron could sustain its own activity without synaptic input by recording from individual neurons from anterodorsal thalamus in brain slices with synaptic blockers. We found that the rebound firing induced by hyperpolarizing pulses often decayed slowly so that a thalamic neuron could keep on firing for many minutes after stimulation. The hyperpolarization-induced persistent firing rate was graded under repeated current injections, and could be enhanced by serotonin. The effect of depolarizing pulses was much weaker and only slightly accelerated the decay of the hyperpolarization-induced persistent firing. Our finding provides the first direct evidence for single-cell persistent activity in the thalamus, supporting the notion that cellular mechanisms at the slow time scale of minutes might potentially contribute to the operations of the head-direction system.

Summary Sparse coding has long been recognized as a primary goal of image transformation in the visual system [1–4]. Sparse coding in early visual cortex is achieved by abstracting local oriented spatial frequencies [5] and by excitatory/inhibitory surround modulation [6]. Object responses are thought to be sparse at subsequent processing stages [7, 8], but neural mechanisms for higher-level sparsification are not known. Here, convergent results from macaque area V4 neural recording and simulated V4 populations trained on natural object contours suggest that sparse coding is achieved in midlevel visual cortex by emphasizing representation of acute convex and concave curvature. We studied 165 V4 neurons with a random, adaptive stimulus strategy to minimize bias and explore an unlimited range of contour shapes. V4 responses were strongly weighted toward contours containing acute convex or concave curvature. In contrast, the tuning distribution in nonsparse simulated V4 populations was strongly weighted toward low curvature. But as sparseness constraints increased, the simulated tuning distribution shifted progressively toward more acute convex and concave curvature, matching the neural recording results. These findings indicate a sparse object coding scheme in midlevel visual cortex based on uncommon but diagnostic regions of acute contour curvature.

We previously demonstrated that it is possible to learn position-independent responses to rotation and dilation by filtering rotations and dilations with different centers through an input layer with MT-like speed and direction tuning curves and connecting them to an MST-like layer with simple Hebbian synapses (Sereno and Sereno 1991). By analyzing an idealized version of the network with broader, sinusoidal direction-tuning and linear speed-tuning, we show analytically that a Hebb rule trained with arbitrary rotation, dilation/contraction, and translation velocity fields yields units with weight fields that are a rotation plus a dilation or contraction field, and whose responses to a rotating or dilating/contracting disk are exactly position independent. Differences between the performance of this idealized model and our original model (and real MST neurons) are discussed.

Background/Aims: To describe a case of left thyroid dysgenesis, accompanied by ipsilateral double parathyroid adenomas in a setting of primary hyperparathyroidism, and to review the pertinent literature on the diagnosis of these rare clinical scenarios. Methods: Review of the English literature with addition of a case report. Results: Preoperative evaluation included both sestamibi and ultrasound evaluation of the neck. Fine-needle aspiration biopsies of what was thought to be two concerning thyroid nodules revealed potential double intrathyroidal parathyroid adenomas. Video-assisted exploration verified double parathyroid adenomas and revealed concomitant left thyroid lobe dysgenesis. Intact parathyroid hormone level returned to normal and a greater than 50% drop from baseline was achieved intraoperatively with subsequent long-term cure. Conclusions: Thyroid dysgenesis is a rare, poorly understood and potentially confusing variety of developmental anomalies, which can be associated with thyroid as well as parathyroid disease. Clinical diagnosis is highly dependent upon the clinician maintaining an index of suspicion for these anomalies, thorough physical examination and careful review of available imaging modalities, especially while investigating thyroid and parathyroid disorders.

This paper is concerned with the dynamical basis of Maxwell's demon within the framework of classical mechanics. We show that the operation of the demon, whose effect is equivalent to exerting a velocity-dependent force on the gas molecules, can be modeled as a suitable force field without disobeying any laws in classical mechanics. An essential requirement for the models is that the phase-space volume should be noninvariant during time evolution. The necessity of the requirement can be established under general conditions by showing that (1) a mechanical device is able to violate the second law of thermodynamics if and only if it can be used to generate and sustain a robust momentum flow inside an isolated system, and (2) no systems with invariant phase volume are able to support such a flow. The invariance of phase volume appears as an independent factor responsible for the validity of the second law of thermodynamics. When this requirement is removed, explicit mechanical models of Maxwell's demon can exist.

It is generally unknown when distinct neural networks having different synaptic weights and thresholds implement identical input-output transformations. Determining the exact conditions for structurally distinct yet functionally equivalent networks may shed light on the theoretical constraints on how diverse neural circuits might develop and be maintained to serve identical functions. Such consideration also imposes practical limits on our ability to uniquely infer the structure of underlying neural circuits from stimulus-response measurements. We introduce a biologically inspired mathematical method for determining when the structure of a neural network can be perturbed gradually while preserving functionality. We show that for common three-layer networks with convergent and nondegenerate connection weights, this is possible only when the hidden unit gains are power functions, exponentials, or logarithmic functions, which are known to approximate the gains seen in some biological neurons. For practical applications, our numerical simulations with finite and noisy data show that continuous confounding of parameters due to network functional equivalence tends to occur approximately even when the gain function is not one of the aforementioned three types, suggesting that our analytical results are applicable to more general situations and may help identify a common source of parameter variability in neural network modeling.

As a rat navigates through a familiar environment, its posi- tion in space is encoded by firing rates of place cells and grid cells. Oscil- latory interference models propose that this positional firing rate code is derived from a phase code, which stores the rat's position as a pattern of phase angles between velocity-modulated theta oscillations. Here we describe a three-stage network model, which formalizes the computational steps that are necessary for converting phase-coded position signals (repre- sented by theta oscillations) into rate-coded position signals (represented by grid cells and place cells). The first stage of the model proposes that the phase-coded position signal is stored and updated by a bank of ring attractors, like those that have previously been hypothesized to perform angular path integration in the head-direction cell system. We show ana- lytically how ring attractors can serve as central pattern generators for producing velocity-modulated theta oscillations, and we propose that such ring attractors may reside in subcortical areas where hippocampal theta rhythm is known to originate. In the second stage of the model, grid fields are formed by oscillatory interference between theta cells residing in dif- ferent (but not the same) ring attractors. The model's third stage assumes that hippocampal neurons generate Gaussian place fields by computing weighted sums of inputs from a basis set of many grid fields. Here we show that under this assumption, the spatial frequency spectrum of the Gaussian place field defines the vertex spacings of grid cells that must provide input to the place cell. This analysis generates a testable predic- tion that grid cells with large vertex spacings should send projections to the entire hippocampus, whereas grid cells with smaller vertex spacings may project more selectively to the dorsal hippocampus, where place fields are smallest. V C 2008 Wiley-Liss, Inc.

Identifying the optimal stimuli for a sensory neuron is often a difficult process involving trial and error. By analyzing the relationship between stimuli and responses in feedforward and stable recurrent neural network models, we find that the stimulus yielding the maximum firing rate response always lies on the topological boundary of the collection of all allowable stimuli, provided that individual neurons have increasing input-output relations or gain functions and that the synaptic connections are convergent between layers with nondegenerate weight matrices. This result suggests that in neurophysiological experiments under these conditions, only stimuli on the boundary need to be tested in order to maximize the response, thereby potentially reducing the number of trials needed for finding the most effective stimuli. Even when the gain functions allow firing rate cutoff or saturation, a peak still cannot exist in the stimulus-response relation in the sense that moving away from the optimum stimulus always reduces the response. We further demonstrate that the condition for nondegenerate synaptic connections also implies that proper stimuli can independently perturb the activities of all neurons in the same layer. One example of this type of manipulation is changing the activity of a single neuron in a given processing layer while keeping that of all others constant. Such stimulus perturbations might help experimentally isolate the interactions of selected neurons within a network.

It is known that Hebbian synapses, with appropriate weight normalization, extract the first principal component of the input patterns. Anti-Hebb rules have been used in combination with Hebb rules to extract additional principal components or generate sparse codes. Here we show that the simple anti-Hebbian synapses alone can support an important computational function: solving simultaneous linear equations. During repetitive learning with a simple anti-Hebb rule, the weights onto an output unit always converge to the exact solution of the linear equations whose coefficients correspond to the input patterns and whose constant terms correspond to the biases, provided that the solution exists. If there are more equations than unknowns and no solution exists, the weights approach the values obtained by using the Moore-Penrose generalized inverse (pseudoinverse). No explicit matrix inversion is involved and there is no need to normalize weights. Mathematically, the anti-Hebb rule may be regarded as an iterative algorithm for learning a special case of the linear associative mapping. Since solving systems of linear equations is a very basic computational problem to which many other problems are often reduced, our interpretation suggests a potentially general computational role for the anti-Hebbian synapses and a certain type of long-term depression (LTD).

Publisher Summary This chapter discusses the accuracy and learning in neuronal populations. The information about various sensory and motor variables contained in neuronal spike trains is quantified by either Shannon mutual information or Fisher information. The accuracy of encoding and decoding by a population of neurons as described by Fisher information has some general properties, including a universal scaling law with respect to the width of the tuning functions. The theoretical accuracy for reading information from population activity can be reached, in principle, by Bayesian reconstruction, which can be simplified by exploiting Poisson spike statistics. The Bayesian method can be implemented by a feedforward network, where the desired synaptic strength can be established by a Hebbian learning rule that is proportional to the logarithm of the pre-synaptic firing rate, suggesting that the method might be potentially relevant to biological systems.

Zhang, Kechen, Iris Ginzburg, Bruce L. McNaughton, and Terrence J. Sejnowski. Interpreting neuronal population activity by reconstruction: unified framework with application to hippocampal place cells. J. Neurophysiol. 79: 1017–1044, 1998. Physical variables such as the orientation of a line in the visual field or the location of the body in space are coded as activity levels in populations of neurons. Reconstruction or decoding is an inverse problem in which the physical variables are estimated from observed neural activity. Reconstruction is useful first in quantifying how much information about the physical variables is present in the population and, second, in providing insight into how the brain might use distributed representations in solving related computational problems such as visual object recognition and spatial navigation. Two classes of reconstruction methods, namely, probabilistic or Bayesian methods and basis function methods, are discussed. They include important existing methods as special cases, such as population vector coding, optimal linear estimation, and template matching. As a representative example for the reconstruction problem, different methods were applied to multi-electrode spike train data from hippocampal place cells in freely moving rats. The reconstruction accuracy of the trajectories of the rats was compared for the different methods. Bayesian methods were especially accurate when a continuity constraint was enforced, and the best errors were within a factor of two of the information-theoretic limit on how accurate any reconstruction can be and were comparable with the intrinsic experimental errors in position tracking. In addition, the reconstruction analysis uncovered some interesting aspects of place cell activity, such as the tendency for erratic jumps of the reconstructed trajectory when the animal stopped running. In general, the theoretical values of the minimal achievable reconstruction errors quantify how accurately a physical variable is encoded in the neuronal population in the sense of mean square error, regardless of the method used for reading out the information. One related result is that the theoretical accuracy is independent of the width of the Gaussian tuning function only in two dimensions. Finally, all the reconstruction methods considered in this paper can be implemented by a unified neural network architecture, which the brain feasibly could use to solve related problems.

Although Papez circuit was initially proposed as a neural substrate for emotions, recent findings of head-direction cells demonstrate that some parts of the structure are involved with representation of spatial orientation. We discuss several problems related to the dynamics of the spatial representation, including (1) reduction of the dynamics of coupled cell groups to that of a simple cell group, in which the formulation of the shift mechanism has been determined uniquely, (2) the relation between dynamic behaviors of coupled cell groups and lesion results, and (3) the limitation of one-dimensional spatial representation in geometric phase problem.

The head-direction (HD) cells found in the limbic system in freely mov ing rats represent the instantaneous head direction of the animal in the horizontal plane regardless of the location of the animal. The internal direction represented by these cells uses both self-motion information for inertially based updating and familiar visual landmarks for calibration. Here, a model of the dynamics of the HD cell ensemble is presented. The stability of a localized static activity profile in the network and a dynamic shift mechanism are explained naturally by synaptic weight distribution components with even and odd symmetry, respectively. Under symmetric weights or symmetric reciprocal connections, a stable activity profile close to the known directional tuning curves will emerge. By adding a slight asymmetry to the weights, the activity profile will shift continuously without disturbances to its shape, and the shift speed can be controlled accurately by the strength of the odd-weight component. The generic formulation of the shift mechanism is determined uniquely within the current theoretical framework. The attractor dynamics of the system ensures modality- independence of the internal representation and facilitates the correction for cumulative error by the putative local-view detectors. The model offers a specific one-dimensional example of a computational mechanism in which a truly world-centered representation can be derived from observer-centered sensory inputs by integrating self-motion information.

SummaryAs a consequence of conditioning visual cues with delayed reward, cue-evoked neural activity that predicts the time of expected future reward emerges in the primary visual cortex (V1). We hypothesized that this reward-timing activity is engendered by a reinforcement signal conveying reward acquisition to V1. In lieu of behavioral conditioning, we assessed in vivo whether selective activation of either basal forebrain (BF) or cholinergic innervation is sufficient to condition cued interval-timing activity. Substituting for actual reward, optogenetic activation of BF or cholinergic input within V1 at fixed delays following visual stimulation entrains neural responses that mimic behaviorally conditioned reward-timing activity. Optogenetically conditioned neural responses express cue-evoked temporal intervals that correspond to the conditioning intervals, are bidirectionally modifiable, display experience-dependent refinement, and exhibit a scale invariance to the encoded delay. Our results demonstrate that the activation of BF or cholinergic input within V1 is sufficient to encode cued interval-timing activity and indicate that V1 itself is a substrate for associative learning that may inform the timing of visually cued behaviors.