A scaling law derived from optimal dendritic wiring

Abstract

The wide diversity of dendritic trees is one of the most striking features of neural circuits. Here we develop a general quantitative theory relating the total length of dendritic wiring to the number of branch points and synapses. We show that optimal wiring predicts a 2/3 power law between these measures. We demonstrate that the theory is consistent with data from a wide variety of neurons across many different species and helps define the computational compartments in dendritic trees. Our results imply fundamentally distinct design principles for dendritic arbors compared with vascular, bronchial, and botanical trees.computational neuroscience | branching | dendrite | morphology | minimum spanning tree O ne of the main roles of dendrites is to connect a neuron to its synaptic inputs. To interpret neural connectivity from morphological data, it is important to understand the relationship between dendrite shape and synaptic input distribution (1-4). As early as the end of the 19th century (5), it was suggested that dendrites optimize connectivity in terms of cable length and conduction time costs, and a number of recent studies have supported the idea that optimal wiring explains dendritic branching patterns using simulations (6-8) or by reasoning from first principles (1, 2, 9, 10). However, although dendrite length is the most common measure for molecular studies of dendritic growth (11), its relationship to dendritic branching and the number of synaptic contacts has not been elucidated. Understanding this relationship should provide crucial constraints for circuit structure and function. Here we directly test the hypothesis that neurons wire up a space in an optimal way by studying the consequences for dendrite length and branching complexity. We derive a simple equation that directly relates dendrite length with the number of branch points, dendrite spanning volume, and number of synapses. ResultsRelating Total Dendritic Length to Optimal Wiring. We assume that a dendritic tree of total length L connects n target points distributed over a volume V (Fig. 1A). Each target point occupies an average volume V =n. A tree that optimizes wiring will tend to connect points to their nearest neighbors, which are on average located at distances proportional to ðV =nÞ 1=3 . We need at least n such dendritic sections to make up the tree. The total length L of these sections sums up toThis result shows that a 2/3 power law relationship between L and n (12) provides a lower bound for the total dendritic length, where c is a proportionality constant. Approximating the volume around each target point by a sphere, then c ¼ ð3=4πÞ 1=3 , and each dendritic section corresponds to the radius of a sphere, giving(SI Text and Fig. S1). Importantly, assuming a constant ratio between the number of branch points, bp and the number of target points (which is addressed later), this assumption also results in a 2/3 power law between wiring length and the number of branch points. Supporting these intuitive derivations of power laws, there ...

Causal Groupoid Symmetries and Big Data

“…Neuroscientific insight at that time was that dendritic trees obeyed a 4/3 power law, which is not optimal. However, in 2012, a team of neuroscientists working independently found the 4/3 power law to be incorrect, and proposed instead a 2/3 power law that is optimally short, based on extensive experimental evidence from humans and other species [5]. This confirmed the prediction.…”

“…Block tunneling with block as-is and block inversion followed with city tunneling to restore the city balance before a decision is made regarding which tours to keep and which to discard. Scales as ( ) 5 O N . 6) fbp-In-place full block permutations.…”

Black-box Brain Experiments, Causal Mathematical Logic, and the Thermodynamics of Intelligence

“…This consideration has motivated SP to propose that AGI and brain function can, and must be explained from and only from the fundamental properties of matter (Pissanetzky, 2011a. A strong confirmation of this point of view was provided by a team of neuroscientists (Cuntz, Mathy, and Häusser, June 2012) who in 2012 independently discovered that dendritic trees in the human and animal brain must satisfy a 2/3 power law that makes them optimally short. SP had predicted in 2011 that dendritic trees had to be optimally short in order for CML to work in the brain, irrespective of what caused them to be optimally short, and proposed that they were optimally short because evolution optimized space and biological resources in the brain, and that CML followed as a side effect.…”

“…Critical to add credibility to our proposal is to verify that the dendrites are optimally short. They are, indeed (Cuntz, Mathy, and Häusser, June 2012).…”

“…The discovery of the causal theory, followed with the prediction in 2011 of a major feature of the brain, that dendritic trees are optimally short (Pissanetzky, 2011a, § 3.8), directly from the properties of information, and its subsequent independent experimental confirmation in 2012 with a wide experimental base including humans and other species (Cuntz, Mathy, and Häusser, June 2012), have shed additional light on the thesis and brought it to a point where it can be used for brain research.…”