Cortex: Statistics and Geometry of Neuronal Connectivity

“axons from single time point in vivo imaging sessions as well as postmortem in slices and found that our method produced interbouton distance distributions that were highly consistent across the two data sets (Figure 2) and that were similar to previously published findings (Amir et al., 1993; Anderson et al., 2003; B…”

“The prevailing view holds that for excitatory neurons, this overlap directly defines synaptic circuits: where axons and dendrites are sufficiently close, synaptic connections occu”

“Assuming neuronal density of 20,000–100,000 cells/mm3 (Braitenberg and Schüz 1998; West et al. 1991), the illuminated volume contained an estimated number of 20–100 cells, many of which can be monitored by a multisite shank of a silicon probe (Buzsáki 2004).”

“Synaptic conductance values:ge = 30 nS,gi = 30 nS Small-size networks of TC and RE cells were considered by considering more diffuse and random connectivity between the two TC and RE layers, and weaker synaptic weights (see scheme in Fig.”

“The density of neurons under cortical surface in various mammalian species is ∼ 100,000/mm 2 (Braitenberg and Schüz, 1998), which translates to a span of ∼ 10 μm 2 per neuron.”

“…2000), and 24,000–80,000 for human cortex (Abeles 1991)], it is much smaller than the total number of neurons in the brain [1.6 107 for mouse (Braitenberg and Schüz 1998), 1010 for humans (Abeles 1991)], even compared with the number of neurons that would be contained in the volume of the…”

“Each pyramidal neuron in mouse neocortex receives 10,000 –15,000 excitatory inputs (Braitenberg and Schüz, 1998).”

“This assumption is certainly valid within a macroscopic app roach, and justified by the fact that the neuropil of cerebral cortex is made of a quasi-random arrang ement of cellular processes of very diverse size (8).”

“The overall innervation pattern depends on the manner of axonal outgrowth and branching and on the arrangement of presynaptic specializations along the axon (Hellwig et al., 1994; Braitenberg and Schütz, 1998; Anderson et al., 2002).”

“Statistical properties of the connectivities have been studied in details for instance in [8].”

“This can be related to biochemical and even structural changes in the neurons, for example to growth and modification of dendritic spines (Braitenberg and Schüz, 1998; Engert and Bonhoeffer, 1999).”

“circuitry is primarily local and serves to connect cells with similar receptive fields (Braitenberg and Schüz, 1998; Ko et al., 2011).”

“Cortical neurons are imbedded in a richly interconnected network and receive tens of thousands of synaptic inputs (reviewed in White, 1989; Braitenburg and Shuz, 1998).”

“They are characterized as nonlinear, all-or-none dendritic responses which can propagate actively for some distance and are often confined within the generating branch (Antic et al., 2010; Larkum and Zhu, 2002; Schiller et al., 1997, 2000b).”

“Glutamate Glutamate is the most common neurotransmitter in the mammalian and teleost brain, with an expected 80%–90% of mammalian synapses using glutamate as the neurotransmitter [31].”

“6616 ), and brain regions with high magnitude values are highly alike (greater than 60% overlap) to the visual and sensorimotor cortices [41].”

“Typical values are 100100 pixels, covering from about 3 3 mm to 3 3 cm of cortical tissue, which gives about 5 to 5000 neurons per pixel in superficial layers according to neuronal densities published previously (Braitenberg & Schuz, 1998 ).”

“Based on the statistics of cortical connections, it has been argued that the cortex is, in fact, an auto-associative memory (Braitenberg & Schüz, 1998) and a network with similar auto-rich auto-associative connectivity (Palm, 1993; Willshaw, Buneman, & Longuet-Higgins, 1969) was therefore chosen as…”

“Without such restriction, they would be expected to appear in the same proportion in which they are present in the network at large ( 1– 4) (Braitenberg and Schüz, 1998).”

“The fractional area of the entire axon amounts to 30% (Braitenberg et al., 1998).”

“Synapses can occur where axonal and dendritic branches of different neurons come sufficiently close to each other (Braitenberg and Schütz 1998; Peters 1979).”

“Furthermore, cortical inhibitory neurons and synapses are only one tenth as numerous as excitatory neurons (Abeles, 1991; Braitenberg & Schuz, 1998).”

“[1], [8] and are given by the following expressions (26):”

“Reported values for m0 are in the range of 5–30 (25, 26).”

“…semantic feature units (filled circles at the top) characterizing the related concept. b | Auto-associative networks include connections between their neurons, such that reverberating activity is possible; they are inspired by the local connectivity between adjacent cortical pyramidal cells 20,22 .”

“Cortical neurons are densely innervated, and the majority of their synapses arise from neighboring cortical neurons (Braitenburg and Shuz, 1998).”

“Braitenberg and Schuz, 1998 ), which can be used to compile probabilistic connection matrices.”

“These equations are simplified by imposing the random connectivity approximation, in which the number of synapses within the cortex is proportional to the product of the numbers of sending and receiving neurons (Braitenberg and Schüz, 1998; Robinson et al., 2001a; Wright and Liley, 1995).”

“Even the strongest proponents of statistical connectivity or neurogeometry (22, 23) believe that arbor alignment requires extracellular chemical repulsion (24) and attraction (25).”

“In this section we present the final mathematical formulation of the thalamocortical model proposed, with SRCs and LRCs included.”

“Because the product of the number of neurons and axonal surface area is proportional to the ratio of the volume of gray matter to axon diameter (due to the empirical fact that volumes of intracortical axons and gray matter are proportional across species [28]), the resting potential contribution produces the total met…”

“…synapses between them (Ramo´n y Cajal 1891; Sholl 1956; Szenta´gothai 1975; Gilbert and Wiesel 1981; Gilbert 1983; White and Keller 1989; Abeles 1991; Braitenberg and Schu¨z 1998; Kalisman et al. 2003, 2005; Lu¨bke et al. 2003; Binzegger et al. 2004; Amirikian 2005; Shepherd et al. 2005) (for…”

“A typical cortical neuron receives input from 10,000 other neurons (Braitenberg and Schuz, 1998), distributed within the same cortical column as well as much more distal parts of cortex and elsewhere.”

“the cortex, because the retina is highly ordered, whereas the cortex looks more random (Braitenberg and Schüz, 1998).”

“…18] have shown that it is advantageous to quantify the dynamics by defining an order parameter Csyn(K) as an average over pairs of 4 neurons of the correlations between neurons i, j Ci,j(τ) = 1 τ + 1 t0+τ∑ t=t0 σi(t)σj(t), (5) where σi(t) is one if the neuron i spiked at time t and zero otherwise.”

“It may further indicate that, although local synaptic connectivity is suggested to become dense in adulthood (Braitenberg and Schüz, 1998), functional connectivity orchestrated on top of a subset of structural connectivity remains sparse— keeping the cost of functional information processing low…”

“…“cell assembly” is used in different ways by different researchers, reflecting a degree of variability of the CELL ASSEMBLY concept, it seems appropriate to spell out how we use the term in the present context (see also Braitenberg 1978; Palm 1982; Braitenberg and Schüz 1998; Pulvermüller 2002).”

“Because our model network is smaller than real cortical networks, where each neuron receives inputs from the order of 10,000 neurons (Braitenberg and Schüz, 1998), we added a background current to represent inputs from many neurons, as previously done by (Destexhe et al., 2001).”

“the local cortico–cortical connectivity with wiring densities up to 4 km of axon per mm(3) gray matter [24].”

“The average number of synapses of neurons (*104) (Braitenberg and Schüz 1998) divided by the number of neural elements (*1010) (Herculano-Houzel 2012) results in a very low average probability of any two neurons in the brain making contact (10-6), implying a highly dispersed network.”

“Morphological clustering of all neurons excluded the axon and was based on measurements of the following 11 parameters for each neuron: (1) presence of an axon (binary parameter), (2) fractal dimension, (3) total process length, (4) number of branch nodes, (5) longest neurite length, (6) volume of the convex hull arou…”

“Methods). Then, by using Equation 27 and the constancy of the optimal local wire diameter d across different species [ 31 ], we predict that R0 ; V 4/27 . This prediction agrees well with”

“Some have attempted to resolve this problem by using geometrical methods to estimate “projected spine counts” based on Golgi data (Feldman and Peters, 1979; Larkman, 1991; Braitenberg and Schuz, 1998).”

“We denote by α i the related dimensionless coefficients (ie, J i =α i J) that allow experimental fit, following Braitenberg and Schuz ( 1998 ).”

“Given that over 80% of neurons are excitatory and over 90% of synapses release glutamate, the major player in neuroenergetics is glutamatergic activity [174–176].”

“This may be explained by the fact that glutamatergic neurons constitute the vast majority of the neuronal population (Braitenberg and Schüz 1998), hence general elevation of brain glutamine levels will not only support GABAergic neurons but even more so the dominant population of excitatory…”

“At this spatial scale, pyramidal-to-pyramidal connections become increasingly important (Braitenberg and Schüz, 1998).”

“synapses formed by glutamatergic terminals) in the cortex are formed almost exclusively on dendritic spines [59], and this amount is similar to the magnitude of reduction in vGluT1-immunoreactive profiles observed in this study, one interpretation of our finding is that the majority of the glutamatergic terminals that…”

“This produces a sparse, patchy and topographic connectivity, as typically found in the mammalian cortex (Amir et al., 1993; Braitenberg and Schüz, 1998; Douglas and Martin, 2004; Kaas, 1997).”

“In mammals, where 1 mm3 of cortex contains 105 neurons, 108 synapses and 4 km of axons [22], similar combined detailed and exhaustive approaches is clearly not feasible.”

“At this level and the next – that of actual synaptic connections – the debate about specificity has been couched in terms of Peters’ rule [19,22,23], that is, axons make connections randomly in direct proportion to the occurrence of all synaptic targets in the adjacent neuropil, with no local specificity.”

“As has been reported by others (Braitenberg and Schuz, 1998), tissues could show alterations from 42% to 80% of their original volume due to fixation and staining effects (Drenhaus et al., 1986; Mouritzen Dam, 1979; O'Kusky and Colonnier, 1982; Romeis, 1968; Schuz and Palm, 1989; Stephan, 1960;…”

“Concerning inhibition, thisislessclear.Thedensityofcorticalinhibitoryneuronsis 10–15 times lower than excitatory neurons, which may imply that inhibitory neurons contribute less to the overall BOLD response [ 36 ].”

“This negative feedback circuit functions both as a ‘local’ activation control and lateral inhibition mechanism, simulating the action of a pool of inhibitory interneurones surrounding a pyramidal cell in the cortex (Braitenberg & Schu¨z, 1998).”

“…into account (Husain et al., 2004; Guenther et al., 2006; Ueno et al., 2011), they either did not incorporate learning mechanisms, or made use of ones (e.g. back-propagation) whose neurobiological plausibility is questionable (Mazzoni et al., 1991; Braitenberg & Sch€uz, 1998; O’Reilly, 1998).”

“It is pertinent to note that in cortex the number of axons and dendrites are approximately equal (Braitenberg and Schüz, 1998c; a), and that dendritic structures change with age and can vary between cortical areas (Jacobs and Scheibel, 2002).”

“and dendrites elongate, branch and form synapses should be investigated (Abeles, 1991; Braitenberg and Schütz, 1998; Anderson et al., 2002; Thomson and Morris, 2002).”

“What mechanisms produce these neural rhythms? The complexity of the vertebrate brain—resulting not only from the sheer number of neurons (approximately 10(9)) and their connections (approximately 10(11) [4]), but also from the many different neuron classes (e.”

“…in which extensive associative memory can be established by synaptic strengthening between adjacent neurons and/or neurons in distant brain areas (Braitenberg and Schüz, 1998); and (iii) repeated synchronous neuronal activation and subsequent synaptic strengthening has important functional…”

“, 2000) and brain connectivity is also surprisingly similar at various points (Henderson and Robinson, 2013, 2014; Braitenberg and Schüz, 1991).”

“This accounts for the sparseness of basal dendritic arbors of pyramidal cells, which form potential synapses with only a small fraction of axons passing through the arbor territory (6) (Fig.”

“In cortical gray matter, histology studies in mice have reported a rather even split between axons and dendrites and a neurite volume fraction of approximately 60% (Braitenberg & Schüz, 1998; Chklovskii, Schikorski, & Stevens, 2002; Ikari & Hayashi, 1981).”