Cortex: Statistics and Geometry of Neuronal Connectivity

“Braintenberg and Schuz found that only 1 in 100 or even 1000 white matter fibers connect the cortex with subcortical areas; inter-and intra- hemispheric cortical projections constitute the rest of the white matter fibers (Braitenberg et al., 1998).”

“In (13–15) we have assumed the approximation of cortical connectivity (Braitenberg and Schüz 1998; Robinson et al. 1998; Wright and Liley 1996) where the number of projections from each source neuron to each target population is proportional the target population’s size.”

“This number is consistent with averages calculated across cortical layers and areas in mice (Braitenberg and Schüz, 2013), cats (Beaulieu and Colonnier, 1989) and humans (Pakkenberg et al., 2003).”

“We will consider connectivities of 10%, 20% and 30% (α = 0.1, 0.2 and 0.3) in agreement with typical values in the brain (Braitenberg and Schüz 1998).”

“Some symmetries are widely considered biologically plausible for cortical maps, for instance the invariance under spatial translations and rotations or a global shift of orientation preference [51,63,67–71].”

“…matches that seen in the brain (Nunez, 1995; Robinson et al., 1997, 2016; Nunez et al., 2001; Mukta et al., 2019, 2020) and yields eigenfunctions that are very similar to those obtained from connectivity matrices and anatomical studies (Robinson et al., 1997, 2016; Braitenberg and Schüz, 1998).”

“The relationship between the connection of two neurons and the overlap of their arborizations can be described by Peters’ rule [7].”

“It is in accordance with anatomical histology which shows even more short fibers ( < 10 mm; Braitenberg and Schuz, 1998) and neural development mechanism promoting short wiring (Sporns, 2010).”

“Every cortical pyramidal cell receives approximately 10,000 synaptic inputs, about 75% of which are excitatory, the exact numbers varying between structures and species (Abeles, 1991; Braitenberg and Schüz, 1998).”

“However, 22 computational properties of neural systems are also dependent on the structure and dynamics in the neural dendrites of 23 single neurons [13], which make up the vast majority of neural tissue [14].”

“The probability of a synapse to be created between any two cells falls off with their distance (Braitenberg and Schüz 1998) according to a Gaussian function clipped to 0 outside the chosen neighborhood (a square of size n = 19 for excitatory and n = 5 for inhibitory cell projections).”

“…neurons of the same pool. might be thought of a rather rough hypothesis has been proved to be a direct consequence of the central limit theorem (Amit and Tsodyks, 1991), which well applies to the large number of connections on the dendritic trees of cortical neurons (Braitenberg and Schüz, 1998).”

“The overlap of dendrites and axons gives an estimate of synaptic connectivity; this is called Peters’s rule (Peters and Feldman, 1976; Braitenberg and Schüz, 1998).”

“If intracortical connectivities are proportional to the numbers of synapses involved (Braitenberg and Schüz 1998), one finds Vi = Ve and Qi = Qe (Robinson et al. 1997; Wright and Liley 1996), which lets us concentrate on excitatory quantities (inhibitory effects are retained and can be calculated…”

“Roughly 90 percent of all cortical synapses are excitatory and 10 percent are inhibitory (Abeles 1991; Braitenberg and Schüz 1998).”

“Following [34], each Ci is proportional to the maximum number C of synapses between two populations.”

“Yet it is ceaselessly active at all times in responding to input from an average 10,000 other neurons (Braitenberg and Schüz, 1998) by which it is modulated through multiple order parameters.”

“Neuronal synchronization was shown to strengthen synaptic transmission (Braitenberg & Schüz, 1998) and, in line with this, efficacy of motor integration depended on the strength of neuronal synchronization in remote functional neuronal networks (von Stein & Sarnthein, 2000; Hummel & Gerloff, 2006).”

“Each pyramidal neuron receives btween 5000 and 60000 synaptic contacts and a large part of this connectivity originates from the cortex itself (DeFelipe and Farinas, 1992; Braitenberg and Schüz, 1998).”

“Different models have been developed to systematically study structural connectivity (Uttley, 1956; Liley and Wright, 1994; Braitenberg and Schuz, 1998; Stepanyants et al., 2002; Kalisman et al., 2003).”

“One possible explanation for this difference is that a local volume of neuropil integrates numerous synaptic processes (Braitenberg and Shchuz, 1998; Chklovskii et al., 2002) that may extend over considerable cortical distances, providing the same signal to different neuropil patches.”

“Peters’ rule, a method proposed to predict synaptic connectivity on the basis of geometric overlaps between axon and dendrite, has been interpreted to provide support for random synaptic connectivity (Peters and Feldman, 1976; Braitenberg and Schüz, 1998).”

“…onto cortical neurons as these numbers are part of the puzzle in determining the strength of synaptic integration from different sources (feedforward and feedback) and hence functional spiking properties of the different neuronal populations (Braitenberg and Schüz 1998; Oberlaender et al. 2012).”

“For PCs, we reproduced the V1 canonical circuit, 4-6,53 but we found surprising differences for BCs and MCs. Finally, E / E pathways were weaker than E / I pathways.”

“The probability of finding a connected pair is so low even at short distances (Braitenberg and Schüz, 1998) that existing electrophysiological techniques would seem to be inappropriate for searching for interconnected pairs at larger distances.”

“…hundreds to thousands of other pyramidal cells located near and far in the cortical expanse (for details, see Peters, 1987a,b; Douglas and Martin, 1991; Beaulieu et al., 1992; Braitenberg and Schüz, 1998; Thomson and Lamy, 2007; Harris and Mrsic-Flogel, 2013; Callaway and Luo, 2015, pp. 8982–8983).”

“The average number of synapses per neuron is roughly constant at 8000 in cortices ranging in size from mouse to human (Braitenberg & Schuz, 1998; Beaulieu & Colonnier, 1989; Pakkenberg et al., 2003).”

“181, 185 & 186 for ˆ λ † = ˆ λ ∗ ).”

“In addition, the majority of anatomical connections impinging on any cortical neuron are of local origin with distances at the mm scale (Braitenberg and Shuez, 1998).”

“For the purpose of this paper we concentrate on models of the cortex where anatomical data (Beaulieu and Colonnier, 1989; Pakkenberg et al., 2003; Braitenberg and Schüz, 2013) suggests that, across species, cortical neurons received an average of around 8000 synaptic inputs.”

“The role of excitatory and inhibitory synapses in the brain has been discussed in the literature [43, 44].”

“Interestingly, despite the cortex generally being associated with cell bodies rather than connections 95% of connections in the brain are found in the cortex in the form of short-range connections (Braitenberg and Schüz, 1998).”

“The ratio of 4:1 between excitatory and inhibitory neurons is typical for cortical and hippocampal networks [86].”

“A striking anatomical feature of the cortex is its homogeneity; a few types of different neurons are orderly arranged in a repetitive pattern throughout the cortex [4,23].”

“Braitenberg and Schuz estimated that vessels and cell bodies occupy 13.8% of the mouse cortical volume (Braitenberg and Schuz, 1998).”

“…of synapses on V1 neurons are formed by lateral connections arising from neurons with somata distributed over a wide range of cortical separations (Braitenberg & Schuz, 1998) and modeling supports the propagation of signals in a wave-like manner via polysynaptic pathways (e.g., Jirsa & Haken,…”

“One of the central goals of neuroscience is to understand how the structure of neural circuits underlies the processing of information in the brain, and in recent years a considerable effort has been focused on measuring neural connectivity empirically [Shepherd, Pologruto and Svoboda (2003); Bureau, Shepherd and Svob…”

“INTRODUCTION Glutamate is the most common excitatory neurotransmitter in the brain, estimated to be responsible for signaling at 80% of cortical synapses (Braitenberg and Schüz, 1998).”

“A synaptic connection between excitatory neurons, which are majority in the cortex [38], can potentially be generated in that cortical region where axonal and dendritic trees spatially overlap [71].”

“Atwell and Iadecola (2002) also report excitatory activity and report that inhibitory neurons are much rarer in the cortex than excitatory neurons (Abeles, 1991; Braitenberg and Schüz, 1998).”

“Human cortex contains 50000 neurons per mm3 so that a typical 2 mm isotropic voxel contains about 400000 neurons (Braitenberg & Schüz, 1998); the 1 mm isotropic voxels depicted in Figure 2 contain about 50000 neurons each.”

“Thus, ‘‘probing the network’’ through intracellular Vm activity is possible and constitutes a promising approach, but it will require a continuous effort combining theory, computational models and intracellular physiology.”

“If only pyramidal cells, the most common neuron type in the cortex (Braitenberg & Schüz, 1998), are considered and their interactions through glutamate synapses, it may be appropriate to assume that EPSPs have similar decay slopes.”

“All nodes have roughly the same degree, but the length of edges increases with the level, assuming the obvious threedimensional embedding (see Figure 1.b) and in keeping with the anatomical observations of Braitenberg and Schuz [7].”

“…is ∼2.5 mm while those of a pig is ∼2.1 mm as opposed to 0.8– 1.5 mm in rodents, making the extrapolation of porcine data regarding alcohol induced cortical alterations to humans more representative (Mayhew et al., 1996; Braitenberg and Schüz, 1998; Fischl and Dale, 2000; Vetreno et al., 2017).”

“The connectivity and the retrieval procedure can also affect the asymptotic value, but on a smaller scale (Schwenker et al. 1996; Bosch and Kurfess 1998).”

“In every cubic millimeter of cortical tissue in the order of 106 neurons must be wired appropriately for their respective functions such as the analysis of sensory inputs, the storage of skills and memory, or motor control [ 4 ].”

“Brain region and Node color The brain is roughly divided into 5 cortex regions, including prefrontal cortex, premotor cortex, auditory cortex, somatosensory cortex andvisual cortex [5, 34].”

“It is known that the (visual) cortex is composed more of excitatory neurons than inhibitory neurons ( 52 , 53 ); however, recurrent IPSPs are stronger than recurrent EPSPs to maintain a stable, balanced state in the existence of external input ( 39 ).”

“With regard to glutamate (Glu), almost 85% of all synapses in the brain are glutamatergic, making Glu a central molecule in the brain [3].”

“…according to histological data —showing that, in portions of the cortex, there are about four times more excitatory than inhibitory neurons [14, 15, 16, 17]— the E/I ratio is assumed here to be 4, so that any I neuron receives effective excitatory inputs from 32 surrounding E s and any I neuron…”

“phylogenetically younger) brains, neuroanatomical investigations have revealed intricate networks that link neurons and neuronal populations at different levels of scale, ranging from single cells at the microscopic level, to mesoscopic structures such as columns, and to macroscopic structural units such as nuclei and…”

“Each neuron is connected to only a very small fraction of the other neurons (Braitenberg & Schuz, 1998).”

“In the past decades, single-neuron recordings have been complemented by multineuronal experimental techniques, which have provided quantitative evidence that the cells forming the nervous systems are coupled both structurally [8] and functionally (for a recent review, see [75] and references therein).”

“This gives a ratio of 4:1 excitatory to inhibitory neurons, which is consistent with physiology (Braitenberg & Schuz, 1998).”

“…the context of the underlying neuronal population generating macroscopic brain signals, these assumptions hold: coupling among theneurons is given (Braitenberg and Schüz,1998;Hellwig, 2000) and furthermore empirical relationships betweenmacroscopic aggregate signals to collective network…”

“These are subsequently organized into larger columns variously called just “columns”, “cortical columns”, “hypercolumns”, or “modules.” Braitenberg [ 43 ] postulates two general connectivity systems in cortex: “metric” (high density connections to physically local cells, based on actual two-dimensional layout), and “a…”