Cortex: Statistics and Geometry of Neuronal Connectivity

“The small dimension of organelles ( 1 μm2) within cells, as well as the distance between neighboring cells (∼30 nm on average) [28,29] imply that the capacitance values of the media are necessarily small compared to the membrane capacitance, and thus the purely capacitive effects (without diffusion) are likely to be n…”

“3A) ( 56 ).”

“Briefly, the model consists of a recurrent network of N = 500 leaky integrate-and-fire (LIF) neurons, of which f = 80% are excitatory and 20% are inhibitory [83].”

“About 80% of all neurons in the cerebral cortex use glutamate as a neurotransmitter (Braitenberg and Schuz 1998; Jones 2009) and most of the energy in the brain, including the cerebral cortex, is used to support glutamatergic activity (Attwell and Laughlin 2001).”

“The plane of a coronal section is often close to a true perpendicular section, but this is not accurate in the frontal and occipital poles of the cortex (Braitenberg and Schüz, 1998).”

“Within each model area, there are sparse and random connections between excitatory neurons whose probability decrease with distance, and each excitatory neuron projects to one inhibitory cell which, in turn, projects back to it and its neighbors, so as to capture some aspects of local cortical connectivity (Braitenber…”

“Thus, cortical connectivity is neither merely random nor confined to local couplings; in fact, it is a combination of both (Braitenberg and Schüz, 1998; Stepanyants et al., 2009; Voges et al., 2010a,b).”

“Following Wright and Liley (1996), Robinson et al. (1998) and Braitenberg and Sch¨uz (1998) the intracortical connection strengths are exploited.”

“These effects of the peptide are important, because neuronal dendritic arbors form the bulk of cortical volume [31,57]; thus, changes in dendrites are likely to be reflected as measurable changes in cortical structure and function.”

“The latter threshold was based on the fraction of all neurons estimated to be inhibitory by Braitenberg and Schüz (1998). Computerized image analysis produced estimates of the surface fraction (i.”

“The lower energy cost of individual neurons could be attributed to the effect of colder temperature on metabolism and the occurrence of fewer synapses in ectothermic brains [55], or the lower energy cost of individual neurons could be associated to the smaller number of glia in ectotherms [46,47,56].”

“Increasing the spacing interval, i.e., the time between learning repetitions (Figure 1A, Melton, 1970; Toppino and Bloom, 2002; Kahana and Howard, 2005) improves memory performance.”

“Our aim in this report is to summarize our observations of cortical fields in humans and other animals engaged in intelligent behaviors, and then to describe the dynamics of cortex using concepts adapted from non-equilibrium thermodynamics [2] and quantum field theory [3, 4].”

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

“This ongoing activity, which mainly originates from profuse intracortical synaptic connections (Braitenberg & Schüz, 1998; Destexhe et al. 2003), varies widely in amplitude and frequency as a function of the vigilance state and anaesthetic regime.”

“Since cortical neural networks are overall sparse (e.g., Braitenberg and Schüz, 1998; Holmgren et al., 2003), we may also expect some level of dilution for embedded feed-forward chains.”

“Certain pathways through the network may be favoured by preferred synaptic interactions between the neurons following developmental and learning processes [47–49].”

“However, over recent years, it has been found that, on average, cortical neurons have 8,000 input synapses (Beaulieu and Colonnier, 1989; Pakkenberg et al., 2003; Braitenberg and Schüz, 2013) each receiving spikes at the lower average rate of around 3 Hz (Buzsaki and Mizuseki, 2014).”

“Thus, it would seem rather implausible that the brain uses its storage capacity, arguably its most precious resource [29], to store something that it does not have to retrieve.”

“Braitenberg and Schüz (1998) observed that the maximum radius of basal dendrites is 100 μm, while axons can extend by 500 μm or more.”

“…and area-specific global regulation mechanisms (referred to as local and global inhibition, respectively) (Yuille and Geiger, 1998; Palm et al., 2014; Braitenberg and Sch¨uz, 1998); 5) Single cells’ neurophysiological dynamics, including sigmoid transformation of membrane potentials into neuronal…”

“…derived from these observations were that axons grow randomly across the cortical sheet (i.e., with no particular spatial orientation) and that they indiscriminately form connections once they are close enough to a potential target neuron, a mechanism that has been named Peters’s Rule [82, 83].”

“Given the ubiquity of inhibition in the cortex (∼20% of all cortical neurons are inhibitory, see Braitenberg and Schüz, 1998) and its undeniable role in shaping and stabilizing network dynamics and neuronal excitability, the possible functional implications of dynamic inhibition are of great…”

“Because of tight somatic packing, the principal layer volume directly affects the absolute numbers of principal cells (Boss et al., 1987; Amaral et al., 1990; Braitenberg and Schultz, 1998).”

“This is one order of magnitude smaller than the cross-sectional area of the soma of a typical neuron 25 .”

“It has been argued that the connectivity of cortical microcircuits can be approximated by a random model (Braitenberg and Schüz, 1998), but recent anatomical and physiological studies indicate that at least connection probability decreases with distance (for review, see Boucsein et al., 2011),…”

“As reported in the studies by Schüz and Palm (1989) and Braitenberg and Schüz (1998), this density is 0.91 m 3 (n 3 mice).”

“Of the remaining objects, those with areas above the 15th percentile were used for minicolumn fragment detection, and the smaller objects were classified as interneurons, based upon the fraction of all neurons estimated to be inhibitory by Braitenberg & Schüz (1998).”

“Cortical neurons communicate via recurrent excitatory synaptic connections, with each cell typically receiving excitatory inputs from thousands of other cortical neurons (Braitenberg and Schüz, 1998; Douglas et al., 1995).”

“Eighty-five percent of cortical neurons are pyramidal cells, and these are the cells that dominate these field potentials at the scale [45].”

“This compares favorably to estimates of connection probability from anatomical considerations (0.864; Braitenberg and Schüz, 1998).”

“At the level of individual neurons the connectivity pattern is so complex that only generalised statistics can be produced, see ( Braitenberg, and Schuz, 1998 ) for a fascinating discussion.”

“Indeed extracellular fluid is thought to comprise only 12–25% of the brain’s volume (Peters et al., 1991; Braitenberg and Schuez, 1998).”

“Glutamate is the most common neurotransmitter in the brain, mediating signals at 80% of the synapses in the cortex (Braitenberg and Schüz, 1998).”

“With an estimated average convergence and divergence rate of cortical neurons in the order of 104– 105 inputs and outputs (Braitenberg and Schüz, 1998), realistic models even of small cortical patches require the inclusion of several 10,000s of neurons (Potjans and Diesmann, 2012).”

“Following the approach of previous neural field models, excitatory and inhibitory synapses in the cortex are assumed proportional to the numbers of neurons (Wright and Liley, 1996; Braitenberg and Schüz, 1998).”

“The contrast due to high degree of myelination, well known in the primary visual areas (Braitenberg & Schüz, 1998), has decreased the gray/white matter contrast resulting in missegmentation in FSPGR.”

“6*10(7) and d = 7800 have been given for the mouse (Braitenberg and Schuz, 1998) and of n = 10(10) and d = 2.”

“Each cortical neuron receives inputs from approximately 10 4 other neurons and sends out signals via its synapses to about 10 4 others (Abeles, 1991; Braitenberg and Schüz, 1998).”

“They are comparable in size, with thickish, longish dendrites and Ն1,000 excitatory inputs (Braitenberg and Schüz 1998; Freed and Sterling 1988; Kier et al. 1995; Shadlen and Newsome 1994, 1998); and both have a low maintained discharge (ϳ5 spikes/s) so that spike responses resemble " bursts.”

“4 km of cable compared to 4 km of axons in a typical 1 mm(3) cube of cortex [12], dendrites are important information processors and a relevant target for stimulation.”

“Axons of cortical pyramidal neurons form between 7000 and 8000 synapses onto target neurons (Braitenberg & Schüz, 1998).”

“Although our study was performed using a sample from rodent neuropil, its results primarily depend on two parameters: the mean density of synapses in neuropil, ,1–2 mm(23), and the typical size of synapses, ,300 nm [32,33,34,35,36,38].”

“In every mm3 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 [ 11 ].”

“As amatter of fact, the brain organization can be viewed in gross sense as a number of local subnetworks (graymatter) coupled by long distance connections (whitematter) (Braitenberg and Schuz 1998; Kandel et al. 2000).”

“Early studies indicate that around 90% of the synapses in the CNS are glutamatergic (Abeles 1991; Braitenberg and Schüz 1998), whose activity consumes roughly 80% of the whole brain's energy (Attwell and Laughlin 2001; Howarth et al. 2012).”

“The numerical values of these parameters are: VNa~0:050 V, VK~{0:100 V, V0~{0:065 V (standard values), (1{w)&0:65 [26,27,28,56], gNa,0~3:10 {7 (Vcm(2)) [33], C&3:2:10{6 F/cm(2), and d~0:45:10{4 cm [33].”

“As not much is known about a structured connectivity within the subpopulations of layer 2/3, we assume random connectivity (compare Braitenberg & Schüz, 1998; Maass, Natschläger, & Markram, 2002; Kalisman, Silberberg, & Markram, 2005; Amirikian, 2005; Muresan & Savin, 2007).”

“Although some have argued that the local cortical network is for the most part randomly connected, other than topographic specificity [1], detailed examination of cortical connectivity supports the idea that synaptic connectivity is specific at several different levels.”

“It has been observed that there is 350,000 thalamocortical connections per 11,000,000 neurons in layer 4 of the primary visual cortex (Binzegger et al., 2004) leading to a ratio of 1/30 (Izhikevich and Edelman, 2008).”

“Their effect can be comparably strong because several inh. cells preferably connect to proximal parts of their exc. targets (Braitenberg and Schüz 1998).”

“As expected, this value is already much larger than in mammalian nervous tissue (15%) (33), while for Aplysia it is a lower limit, since smaller cell bodies were not accounted for.”

“The cortex, in general (ignoring species and areal density differences [16]) consists of approximately 10(10) nerve cells, each receiving numerous connections of order 10(3) (up to about 10(4)) [7].”

“The probability of having synaptic contacts is much higher between nearby neurons than between distant neurons (Douglas et al., 1995; Braitenberg and Schüz, 1998).”