Brain Size and Number of Neurons: An Exercise in Synthetic Neuroanatomy

Certain remarkable invariances have long been known in comparative neuroanatomy, such as the proportionality between neuronal density and the inverse of the cubic root of brain volume or that between the square root of brain weight and the cubic root of body weight. Very likely these quantitative relations reflect some general principles of the architecture of neuronal networks. Under the assumption that most of brain volume is due to fibers, we propose four abstract models: I, constant fiber length per neuron; II, fiber length proportionate to brain diameter; III, complete set of connections between all neurons; IV, complete set of connections between compartments each containing the square root of the total number of neurons. Model I conforms well to the cerebellar cortex. Model II yields the observed comparative invariances between number of neurons and brain size. Model III is totally unrealistic, while Model IV is compatible with the volume of the hemispheric white substance in different mammalian species.     

Nets with reciprocity bias

The random net is modified by the introduction of a finite probability that an arbitrarily selected axon is reciprocated. The resulting distribution of convergence orders is compared with the corresponding distribution for random nets (the Poisson distribution). It is shown that for small values of the bias the terms of the distribution near the modal term (within a range equal to the square root of the axon density) at first increase as the bias increases, while the remaining terms decrease. The modal term itself is shown to increase monotonically with the bias throughout the whole range of the bias. In some special cases, the general behavior of the terms is calculated for the whole range of the bias. Some implications are discussed related to the statistical properties of sociograms.     

Steady states in random nets: I

A neural net is taken to consist of a semi-infinite chain of neurons with connections distributed according to a certain probability frequency of the lengths of the axones. If an input of excitation is “fed” into the net from an outside source, the statistical properties of the net determine a certain steady state output. The general functional relation between the input and the output is derived as an integral equation. For a certain type of probability distribution of connections, this equation is reducible to a differential equation. The latter can be solved by elementary methods for the output in terms of the input in general and for the input in terms of the output in special cases.     

Highly nonrandom features of synaptic connectivity in local cortical circuits

How different is local cortical circuitry from a random network? To answer this question, we probed synaptic connections with several hundred simultaneous quadruple whole-cell recordings from layer 5 pyramidal neurons in the rat visual cortex. Analysis of this dataset revealed several nonrandom features in synaptic connectivity. We confirmed previous reports that bidirectional connections are more common than expected in a random network. We found that several highly clustered three-neuron connectivity patterns are overrepresented, suggesting that connections tend to cluster together. We also analyzed synaptic connection strength as defined by the peak excitatory postsynaptic potential amplitude. We found that the distribution of synaptic connection strength differs significantly from the Poisson distribution and can be fitted by a lognormal distribution. Such a distribution has a heavier tail and implies that synaptic weight is concentrated among few synaptic connections. In addition, the strengths of synaptic connections sharing pre- or postsynaptic neurons are correlated, implying that strong connections are even more clustered than the weak ones. Therefore, the local cortical network structure can be viewed as a skeleton of stronger connections in a sea of weaker ones. Such a skeleton is likely to play an important role in network dynamics and should be investigated further.     

Beyond the binary case in random nets

Experiments on random binary, ternary, etc. (P=2, 3,…, 10) switching nets are reported. Behavioral cycle lengths are examined as functions of output variety,P, input connectance,K, and net size,N. Overall, output variety appears an influential, well-behaved net property. Strong, but well-behaved interactions appear among net variables. In high connectance nets, median cycle length grows approx. asP ^N/2. Other factors constant, one-connected nets show the shortest cycles, and connectance effects appear to converge asymptotically aroundN. Data for cycle length as a function of net size suggest a concavity not compatible with the Kauffman “square root law” (Kauffman, 1969). Evidence of a positive relationship between cycle length and run-in length is found in two-input nets; weaker evidence is obtained that in higher connectance nets this relationship becomes negative in sign. The “modular complexity” ofP>2 nets is examined briefly.     

Input output curves of aggregates of “simple counter” neurons

The refractory periods of an aggregate of simple “counter” neurons are assumed distributed according to some probability frequency. The output of the aggregate is computed for rectangular and triangular distributions. In particular, it is shown that the maximum output of an aggregate with any triangular distribution cannot exceed the maximum output of its average neuron by a factor greater than 2 ln 2. This puts an upper bound on the amount of departure from the behavior of the average neuron which an aggregate characterized by a certain type of distribution can show. Next, the aggregate is supposed to be subjected to regularly spaced stimuli. Under these conditions, a single neuron will give a discontinuous output curve. If, however, the refractory periods are distributed according to some frequency, the output curve may be “smoothed out.” A general condition on the distribution is derived which makes the output monotone increasing with the input. The condition is applied to some special cases.     

On the influence of pigment-protein interactions on the energy transfer processes in photosynthetic membrane structures. 2. LH2 complex of Rhodobacter sphaeroides

Low-temperature heterogeneous absorption and circular dichroism spectra of the Rb. sphaeroides LH2 complexes are calculated within the framework of the mini-exciton theory and diagonal static random disorder for the pure electronic transitions of the monomeric Bchl molecules. The coupling of Bchl molecules with the surrounding amino acid residues has been shown to change both the exciton distribution between the pigment molecules in each of the exciton states. The value of the delocalization index depends on the excitation wavelength and varies between 2-6 Bchl molecules. The optical transitions occurring at 780-790 and 820 nm have been found to be strongly mixed so that all Bchl molecules of the LH2 complex predetermine absorption in these spectral regions. On the other hand, absorption at 800 and 850 nm is mainly determined by the cycles of 9 and 18 Bchl molecules, respectively. Thus, the light energy absorbed by the B800 molecules at 800 nm is transferred to the B850 molecules by the interlevel exciton relaxation processes due to the population of the heavily mixed 820-nm exciton levels. The width of the heterogeneous absorption band for the cyclic monomeric aggregate has been shown to decrease as compared with the monomeric absorption band by square root(Ndel) time, where Ndel is the mean number of pigments over which the exciton is delocalized within the excited absorption band.     

Method for the quantitative evaluation of the distribution pattern of nuclei in normal and malignant endometrial epithelia

A quantitative method of evaluating the pattern of distribution of nuclei within a cell cluster was developed and applied to endometrial cytology. The distribution pattern (DP) is mathematically expressed using a "DP index," which can be determined as a product of the square root of the nuclear density, square root of n0, and the mean distance between the two nearest nuclei, r. The DP index has a limited range of decreasing values from 1.074 to 0.5 to 0, indicative of regular, random and aggregated patterns of nuclear distribution, respectively. The estimated DP index was 0.831 +/- 0.031 (mean +/- standard deviation) in the clusters of normal endometrial epithelial cells from 16 nonneoplastic cases, but 0.638 +/- 0.041 in malignant epithelia from 19 cases of well-differentiated adenocarcinoma. The index of normal epithelia was close to 0.877 of the regular hexagonal distribution pattern. Contrary to this, the DP index was significantly smaller in well-differentiated adenocarcinoma (P less than .001), approaching 0.5, the theoretical value of a random distribution pattern. These findings suggest that quantitative analysis of the distribution pattern of nuclei can be a useful aid in the cytodiagnosis of endometrial lesions.     

The development of swimming rhythmicity in post-embryonic Xenopus laevis.

The post-embryonic development of 'fictive' swimming in immobilized Xenopus laevis tadpoles has been examined during the first day of larval life. In Xenopus embryos (stage 37-38; Nieuwkoop & Faber 1956), the rhythmic ventral root activity underlying swimming occurs as single brief (ca. 7 ms) compound impulses on each cycle. However, by stage 42 (about 24 h after hatching), ventral root discharge consists of bursts lasting around 20 ms per cycle. In addition to increased burst duration in each cycle of larval swimming, the range of cycle periods within an episode increases, although mean period values (ca. 70-80 ms) remain similar to those of the younger animal. Consequently, motoneurons at developmental stage 42 are active during swimming for a greater percentage (ca. 25%) of cycle time than at stage 37-38 (ca. 10%). Developmental stage 40 (ca. 12 h post-hatching) is an intermediate stage in rhythm development. Ventral root discharge varies from bursts of 10-20 ms at the start of an episode to embryonic (ca. 7 ms) spikes at the end of an episode. Furthermore, discharge varies from bursts of activity in rostral segments of stage 40 larvae to 7 ms spikes more caudally, as in embryos. The data thus suggest that Xenopus swimming rhythmicity develops relatively rapidly, along a rostrocaudal gradient, and may involve acquisition of multiple spiking in spinal neurons.     

Contributions to the mathematical biophysics of the central nervous system with special reference to learning

A learning theory based on the lowering of thresholds of neurons under certain conditions is applied to two “random net” models. The first, a so-called “ganglion-brain” is characterized by completely random connections of all afferent tracts except certain ones which form the pathways for unconditioned responses. Certain expressions are derived which measure the learning potentiality of the ganglion— in particular, with respect to the number of responses which can be learned (conditioning potential) and the amount of interference between the learned responses (redundance potential). The second model concerns the progressive refinement of a response. The efficiency of learning in this case is reflected in the eventual specificity of the response which, in turn, depends on the modification of the distribution of thresholds associated with the neurons governing the responses. Expressions are derived relating the initial distribution of thresholds, the relative effectiveness of the various responses, and certain other parameters to the final distribution of thresholds. For a particular choice of the effectiveness distribution of responses the progressive sharpening of the threshold curve (i.e., progressive specificity of response) is demonstrated. Some implications of the model with respect to the evolution of nervous systems are discussed.     

A behavioral summary for completely random nets

This paper characterizes the cycle structure of a completely random net. Variables such as number of cycles of a specified length, number of cycles, number of cyclic states and length of cycle are studied. A square array of indicator variables enables conveninent study of moment structure. Additionally, exact and asymptotic distributional results are presented.     

Low-frequency vibrational spectra of some homopolypeptides in the solid state

Low-frequency Raman and far-infrared spectra of polyglycine, poly-L -alanine, and poly-L -valine have been measured. The Raman spectra exhibit an intense band near 100 cm^?1 for these homopolypeptides. Lattice calculations of the polyglycines are used to assign the intense Raman band to a rotary lattice mode. For homopolypeptides in the β conformation, an infrared band is observed whose frequency varies inversely with the square root of the mass of the peptide repeating unit. This infrared band is assigned to the hydrogen bond stretching lattice vibration.     

The influence of cortical feature maps on the encoding of the orientation of a short line

The inhomogeneous distribution of the receptive fields of cortical neurons influences the cortical representation of the orientation of short lines seen in visual images. We construct a model of the response of populations of neurons in the human primary visual cortex by combining realistic response properties of individual neurons and cortical maps of orientation and location preferences. The encoding error, which characterizes the difference between the parameters of a visual stimulus and their cortical representation, is calculated using Fisher information as the square root of the variance of a statistically efficient estimator. The error of encoding orientation varies considerably with the location and orientation of the short line stimulus as modulated by the underlying orientation preference map. The average encoding error depends only weakly on the structure of the orientation preference map and is much smaller than the human error of estimating orientation measured psychophysically. From this comparison we conclude that the actual mechanism of orientation perception does not make efficient use of all the information available in the neuronal responses and that it is the decoding of visual information from neuronal responses that limits psychophysical performance. Action Editor: Terrence Sejnowski     

Principle of neuronal organization of defensive reflexes in mollusks

Spontaneous and evoked synaptic activity of command neurons for the defensive response of spiracle closing were studied by simultaneous intracellular recording of activity of several identified CNS neurons in snails. Comparison of monosynaptic EPSPs in command neurons evoked by discharges of presynaptic neurons with spontaneous synaptic potentials indicated that the central organization of the defensive reflex is in the form of a two-layered neuron net in which each neuron of the afferent layer possesses a local receptive field, but which overlaps with other afferent neurons. Each neuron of the afferent layer is connected with each neuron of the efferent layer by monosynaptic excitatory connections that differ in efficiency (maximal only with one neuron of the efferent layer). Both receptive fields of neurons of the afferent layer and "fields of efficiency of synaptic connections" are distributed according to the normal law. As a result of this organization the neuron net acquires a new quality: The action of different stimuli leads to the appearance of differently located "spatial excitation profiles" of efferent layer neurons even when this action of the stimulus occurs not at the center of the receptive field.Institute of Higher Nervous Activity and Neurophysiology, Academy of Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 16, No. 1, pp. 26–34, January-February, 1984.     

Statistical criteria for using short-term measurements as an index of 24-hour mean arterial pressure in unanesthetized unrestrained dogs

This study assessed the statistical validity of short time-interval measurements as estimators of true 24 hour mean arterial pressure in unanesthetized, unrestrained dogs. 24 hour intra-arterial pressure recordings were obtained using a stable FM telemetry system. The 24 hour pressure measurements approximated a normal distribution whose variance was inversely related to the selected averaging interval. Given the variance of a normal distribution one can calculate the 95% confidence interval for any single random measurement. Conversely the number of random samples necessary to be within a prescribed confidence interval can be determined. In this study, the 95% confidence interval for a single, random 30 minute arterial pressure average was calculated to be 11.2 mmHg. Only 4.8 +/- 1.4% of 480 individual 30 minute arterial pressure measurements fell beyond this confidence interval. These outlying values were distributed throughout the 24 hour period. The data suggest that randomly chosen short time-interval measurements may be a valid index of true 24 hour mean pressure if the average variance of a population is known and confidence intervals are defined.     

Role of GABAA-Mediated Inhibition and Functional Assortment of Synapses onto Individual Layer 4 Neurons in Regulating Plasticity Expression in Visual Cortex

Layer 4 (L4) of primary visual cortex (V1) is the main recipient of thalamocortical fibers from the dorsal lateral geniculate nucleus (LGN_d). Thus, it is considered the main entry point of visual information into the neocortex and the first anatomical opportunity for intracortical visual processing before information leaves L4 and reaches supra- and infragranular cortical layers. The strength of monosynaptic connections from individual L4 excitatory cells onto adjacent L4 cells (unitary connections) is highly malleable, demonstrating that the initial stage of intracortical synaptic transmission of thalamocortical information can be altered by previous activity. However, the inhibitory network within L4 of V1 may act as an internal gate for induction of excitatory synaptic plasticity, thus providing either high fidelity throughput to supragranular layers or transmittal of a modified signal subject to recent activity-dependent plasticity. To evaluate this possibility, we compared the induction of synaptic plasticity using classical extracellular stimulation protocols that recruit a combination of excitatory and inhibitory synapses with stimulation of a single excitatory neuron onto a L4 cell. In order to induce plasticity, we paired pre- and postsynaptic activity (with the onset of postsynaptic spiking leading the presynaptic activation by 10ms) using extracellular stimulation (ECS) in acute slices of primary visual cortex and comparing the outcomes with our previously published results in which an identical protocol was used to induce synaptic plasticity between individual pre- and postsynaptic L4 excitatory neurons. Our results indicate that pairing of ECS with spiking in a L4 neuron fails to induce plasticity in L4-L4 connections if synaptic inhibition is intact. However, application of a similar pairing protocol under GABA_ARs inhibition by bath application of 2μM bicuculline does induce robust synaptic plasticity, long term potentiation (LTP) or long term depression (LTD), similar to our results with pairing of pre- and postsynaptic activation between individual excitatory L4 neurons in which inhibitory connections are not activated. These results are consistent with the well-established observation that inhibition limits the capacity for induction of plasticity at excitatory synapses and that pre- and postsynaptic activation at a fixed time interval can result in a variable range of plasticity outcomes. However, in the current study by virtue of having two sets of experimental data, we have provided a new insight into these processes. By randomly mixing the assorting of individual L4 neurons according to the frequency distribution of the experimentally determined plasticity outcome distribution based on the calculated convergence of multiple individual L4 neurons onto a single postsynaptic L4 neuron, we were able to compare then actual ECS plasticity outcomes to those predicted by randomly mixing individual pairs of neurons. Interestingly, the observed plasticity profiles with ECS cannot account for the random assortment of plasticity behaviors of synaptic connections between individual cell pairs. These results suggest that connections impinging onto a single postsynaptic cell may be grouped according to plasticity states.     

On the constancy of cell division rate in the root meristem

This review examines under what circumstances the rate of cell division among cells of the root meristem is known to vary. First, methods are compared that have been used to quantify cell division rate. These can be grouped as being either cytological, in which the rate of accumulation of cells in a particular phase of the cell cycle is determined based on some form of cytological labeling, or kinematic, in which the rate of cell accumulation is determined from the net movement of cells. Then, evidence is reviewed as to whether cell division rates vary between different tissues or cell types, between different positions in the root, or finally between different environments. The evidence is consistent with cells dividing at a constant rate, and well documented examples where cell division rate changes substantially are rare. The constancy of cell division rate contrasts with the number of dividing cells, which varies extensively, and implies that a major point for cell cycle control is governing the exit from the proliferative state at the basal boundary of the meristem.     

Estimating the 3D Pore Size Distribution of Biopolymer Networks from Directionally Biased Data

The pore size of biopolymer networks governs their mechanical properties and strongly impacts the behavior of embedded cells. Confocal reflection microscopy and second harmonic generation microscopy are widely used to image biopolymer networks; however, both techniques fail to resolve vertically oriented fibers. Here, we describe how such directionally biased data can be used to estimate the network pore size. We first determine the distribution of distances from random points in the fluid phase to the nearest fiber. This distribution follows a Rayleigh distribution, regardless of isotropy and data bias, and is fully described by a single parameter—the characteristic pore size of the network. The bias of the pore size estimate due to the missing fibers can be corrected by multiplication with the square root of the visible network fraction. We experimentally verify the validity of this approach by comparing our estimates with data obtained using confocal fluorescence microscopy, which represents the full structure of the network. As an important application, we investigate the pore size dependence of collagen and fibrin networks on protein concentration. We find that the pore size decreases with the square root of the concentration, consistent with a total fiber length that scales linearly with concentration.     

“Ignition” phenomena in random nets

The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state. Possible connections between this model and the phenomenon of the “preset” response are discussed.