Model of the early development of thalamo-cortical connections and area patterning via signaling molecules

The mammalian cortex is divided into architectonic and functionally distinct areas. There is growing experimental evidence that their emergence and development is controlled by both epigenetic and genetic factors. The latter were recently implicated as dominating the early cortical area specification. In this paper, we present a theoretical model that explicitly considers the genetic factors and that is able to explain several sets of experiments on cortical area regulation involving transcription factors Emx2 and Pax6, and fibroblast growth factor FGF8. The model consists of the dynamics of thalamo- cortical connections modulated by signaling molecules that are regulated genetically, and by axonal competition for neocortical space. The model can make predictions and provides a basic mathematical framework for the early development of the thalamo-cortical connections and area patterning that can be further refined as more experimental facts become known.Comment: brain, model, neural development, cortical area patterning, signaling molecule

Cortical composition hierarchy driven by spine proportion economical maximization or wire volume minimization

The structure and quantitative composition of the cerebral cortex are interrelated with its computational capacity. Empirical data analyzed here indicate a certain hierarchy in local cortical composition. Specifically, neural wire, i.e., axons and dendrites take each about 1/3 of cortical space, spines and glia/astrocytes occupy each about $(1/3)^{2}$, and capillaries around $(1/3)^{4}$. Moreover, data analysis across species reveals that these fractions are roughly brain size independent, which suggests that they could be in some sense optimal and thus important for brain function. Is there any principle that sets them in this invariant way? This study first builds a model of local circuit in which neural wire, spines, astrocytes, and capillaries are mutually coupled elements and are treated within a single mathematical framework. Next, various forms of wire minimization rule (wire length, surface area, volume, or conduction delays) are analyzed, of which, only minimization of wire volume provides realistic results that are very close to the empirical cortical fractions. As an alternative, a new principle called "spine economy maximization" is proposed and investigated, which is associated with maximization of spine proportion in the cortex per spine size that yields equally good but more robust results. Additionally, a combination of wire cost and spine economy notions is considered as a meta-principle, and it is found that this proposition gives only marginally better results than either pure wire volume minimization or pure spine economy maximization, but only if spine economy component dominates. In sum, these results suggest that the new spine economy principle may be important for brain evolutionary design in a broader context.Comment: Theoretical paper on optimization of connections in the brain, with comparison to the dat

Approximate Invariance of Metabolic Energy per Synapse during Development in Mammalian Brains

During mammalian development the cerebral metabolic rate correlates qualitatively with synaptogenesis, and both often exhibit bimodal temporal profiles. Despite these non-monotonic dependencies, it is found based on empirical data for different mammals that regional metabolic rate per synapse is approximately conserved from birth to adulthood for a given species (with a slight deviation from this constancy for human visual and temporal cortices during adolescence). A typical synapse uses about glucose molecules per second in primate cerebral cortex, and about five times of that amount in cat and rat visual cortices. A theoretical model for brain metabolic expenditure is used to estimate synaptic signaling and neural spiking activity during development. It is found that synaptic efficacy is generally inversely correlated with average firing rate, and, additionally, synapses consume a bulk of metabolic energy, roughly during most of the developmental process (except human temporal cortex ). Overall, these results suggest a tight regulation of brain electrical and chemical activities during the formation and consolidation of neural connections. This presumably reflects strong energetic constraints on brain development

Scaling of Brain Metabolism and Blood Flow in Relation to Capillary and Neural Scaling

Brain is one of the most energy demanding organs in mammals, and its total metabolic rate scales with brain volume raised to a power of around 5/6. This value is significantly higher than the more common exponent 3/4 relating whole body resting metabolism with body mass and several other physiological variables in animals and plants. This article investigates the reasons for brain allometric distinction on a level of its microvessels. Based on collected empirical data it is found that regional cerebral blood flow CBF across gray matter scales with cortical volume as , brain capillary diameter increases as , and density of capillary length decreases as . It is predicted that velocity of capillary blood is almost invariant (), capillary transit time scales as , capillary length increases as , and capillary number as , where is typically a small correction for medium and large brains, due to blood viscosity dependence on capillary radius. It is shown that the amount of capillary length and blood flow per cortical neuron are essentially conserved across mammals. These results indicate that geometry and dynamics of global neuro-vascular coupling have a proportionate character. Moreover, cerebral metabolic, hemodynamic, and microvascular variables scale with allometric exponents that are simple multiples of 1/6, rather than 1/4, which suggests that brain metabolism is more similar to the metabolism of aerobic than resting body. Relation of these findings to brain functional imaging studies involving the link between cerebral metabolism and blood flow is also discussed

Systems level circuit model of C. elegans undulatory locomotion: mathematical modeling and molecular genetics

To establish the relationship between locomotory behavior and dynamics of neural circuits in the nematode C. elegans we combined molecular and theoretical approaches. In particular, we quantitatively analyzed the motion of C. elegans with defective synaptic GABA and acetylcholine transmission, defective muscle calcium signaling, and defective muscles and cuticle structures, and compared the data with our systems level circuit model. The major experimental findings are: (i) anterior-to-posterior gradients of body bending flex for almost all strains both for forward and backward motion, and for neuronal mutants, also analogous weak gradients of undulatory frequency, (ii) existence of some form of neuromuscular (stretch receptor) feedback, (iii) invariance of neuromuscular wavelength, (iv) biphasic dependence of frequency on synaptic signaling, and (v) decrease of frequency with increase of the muscle time constant. Based on (i) we hypothesize that the Central Pattern Generator (CPG) is located in the head both for forward and backward motion. Points (i) and (ii) are the starting assumptions for our theoretical model, whose dynamical patterns are qualitatively insensitive to the details of the CPG design if stretch receptor feedback is sufficiently strong and slow. The model reveals that stretch receptor coupling in the body wall is critical for generation of the neuromuscular wave. Our model agrees with our behavioral data(iii), (iv), and (v), and with other pertinent published data, e.g., that frequency is an increasing function of muscle gap-junction coupling.Comment: Neural control of C. elegans motion with genetic perturbation

Thermodynamic constraints on fiber diameter, neural activity, and brain temperature

There have been suggestions that heat caused by cerebral metabolic activity may constrain mammalian brain evolution, architecture, and function [1-4]. This study [5] investigates physical limits on brain wiring and corresponding changes in brain temperature that are imposed by thermodynamics of heat balance determined mainly by Na/KATPase, cerebral blood flow, and heat conduction. It is found that even moderate firing rates cause significant intracellular Na build-up, and the ATP consumption rate associated with pumping out these ions grows nonlinearly with frequency. Surprisingly, the power dissipated by the Na/K pump depends biphasically on frequency, which can lead to the biphasic dependence of brain temperature on frequency as well. Both the total power of sodium pumps and brain temperature diverge for very small fiber diameters, indicating that too thin fibers are not beneficial for thermal balance. For very small brains blood flow is not a sufficient cooling mechanism deep in the brain. The theoretical lower bound on fiber diameter above which brain temperature is in the operational regime is strongly frequency dependent but finite due to synaptic depression. For normal neurophysiological conditions this bound is at least an order of magnitude smaller than average values of empirical fiber diameters, suggesting that mammalian brains operate in the thermodynamically safe regime. Analytical formulas presented can be used to estimate average firing rates in mammals, and relate their changes to changes in brain temperature, which can have important practical applications. In general, activity in larger brains is found to be slower than in smaller brains