Computational capacity of pyramidal neurons in the cerebral cortex
Danko D. Georgiev, Stefan K. Kolev, Eliahu Cohen, James F. Glazebrook
CComputational capacity of pyramidal neurons in the cerebral cortex
Danko D. Georgiev a, ∗ , Stefan K. Kolev b , Eliahu Cohen c , James F. Glazebrook d a Institute for Advanced Study, 30 Vasilaki Papadopulu Str., Varna 9010, Bulgaria b Institute of Electronics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee Blvd., Sofia 1784, Bulgaria c Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat Gan5290002, Israel d Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, IL 61920, USA
Abstract
The electric activities of cortical pyramidal neurons are supported by structurally stable, morphologicallycomplex axo-dendritic trees. Anatomical differences between axons and dendrites in regard to their lengthor caliber reflect the underlying functional specializations, for input or output of neural information, re-spectively. For a proper assessment of the computational capacity of pyramidal neurons, we have analyzedan extensive dataset of three-dimensional digital reconstructions from the NeuroMorpho.Org database, andquantified basic dendritic or axonal morphometric measures in different regions and layers of the mouse, rator human cerebral cortex. Physical estimates of the total number and type of ions involved in neuronal elec-tric spiking based on the obtained morphometric data, combined with energetics of neurotransmitter releaseand signaling fueled by glucose consumed by the active brain, support highly efficient cerebral computationperformed at the thermodynamically allowed Landauer limit for implementation of irreversible logical oper-ations. Individual proton tunneling events in voltage-sensing S4 protein α -helices of Na + , K + or Ca ionchannels are ideally suited to serve as single Landauer elementary logical operations that are then amplifiedby selective ionic currents traversing the open channel pores. This miniaturization of computational gatingallows the execution of over 1.2 zetta logical operations per second in the human cerebral cortex withoutcombusting the brain by the released heat. Keywords: action potential, brain energetics, logical operation, morphometry, pyramidal neuron
1. Introduction
The cerebral cortex is the seat of higher cognitive functions in mammals. Structurally, it is dividedinto neocortex, made up of six layers of neurons, and allocortex, made up of just three or four layers ofneurons (Rockland & DeFelipe, 2018). The neocortex forms the largest, outer layer of the cerebrum. Inlarge mammals and primates, the neocortex is folded into grooves and ridges, which minimize the brainvolume, and are pivotal for the wiring of the brain and its functional organization (Rakic, 2009). Theneocortex is involved in sensory perception, awareness, attention, motor control, working memory, thought,intelligence, and consciousness (Page, 1981). The allocortex includes evolutionary older regions, such asthe olfactory system and the hippocampus, which comprise the neural basis of emotion and play importantroles in time ordering of memorized events or the consolidation of conscious memory from short-term tolong-term memory (Fournier et al., 2015; Squire et al., 2015; Wible, 2013).Excitatory, glutamatergic pyramidal neurons are the principal type of cell comprising over 70% of allcortical neurons (Nieuwenhuys, 1994). Pyramidal neurons, referred to as the “psychic cells” of the brain byRam´on y Cajal (Goldman-Rakic, 2002), are organized in complex neuronal networks, which communicate ∗ Corresponding author
Email addresses: [email protected] (Danko D. Georgiev), [email protected] (Stefan K. Kolev), [email protected] (Eliahu Cohen), [email protected] (James F. Glazebrook)
Preprint submitted to Brain Research July 26, 2020 a r X i v : . [ q - b i o . N C ] S e p y means of electric signals. Wiring of the corresponding neuronal networks requires individual neuronsto support structurally stable, elongated cable-like projections referred to as neurites . Depending on theirfunctional specialization, the neurites could be classified as dendrites, specialized in delivering inputs tothe neuron, or axons, specialized in delivering outputs from the neuron to other neurons (Georgiev, 2017).Dendrites deliver electric signals through activated synapses mainly formed onto spines of the dendritic tree(Eyal et al., 2018). The post-synaptic electric currents propagate passively along the dendrites through anelectrotonic mechanism that summates the electric signals spatially and temporally at the cell body (soma)of the neuron. Axons output electric spikes (action potentials) in an active fashion that consumes largeamounts of biochemical energy in order to propagate the electric signals without attenuation at a distanceto pre-synaptic axonal buttons whose release of neurotransmitter subsequently affects the electric propertiesof dendrites of target neurons.The morphology of neurites is intimately related to their characteristic functional role (Mounier et al.,2015). Dendrites achieve processing of received information through passive and lossy transmission. Conse-quently, the dendrites have shorter lengths and larger diameters in order to compensate for the electrotonicattenuation of currents with distance. Alternatively, axons are required to deliver output signals at largedistances to target neurons through lossless transmission achieved at the expense of biochemical energy. Toreduce energy expenditure, axons are thinner and insulated with myelin sheets. Thus, a detailed study ofneuronal morphology is essential for better understanding of the neuronal hardware behind higher cognitivefunctions.Here, we analyze a dataset of 749 three-dimensional neuronal reconstructions from NeuroMorpho.org 7.8digital archive (Ascoli et al., 2007). Then, with the use of morphometric, electrophysiological and biochemicaldata, we derive an upper bound on the computational capacity of pyramidal neurons in the cerebral cortex.Finally, we conclude with a theoretical discussion on the fundamental limitations imposed by energetics onpossible subneuronal mechanisms for the processing of cognitive information.
2. Results
Pyramidal neurons are located within layers 2, 3, 5 and 6 of the neocortex (Shipp, 2007). The cell body(soma) of pyramidal neurons has the shape of a pyramid with its base facing towards the deeper layers and itsapex towards the superficial layers of the cerebral cortex (Bekkers, 2011). Because the dendrites of pyramidalneurons from layers 2, 3 and 5 reach layer 1, the size and complexity of their dendritic trees increases withthe depth of the neuron within the cortex. In contrast, the dendrites of layer 6 neurons reach only layer 4,which explain why their dendritic trees are smaller and less complex than layer 5 neurons (Figure 1). Onaverage across all types of cortical pyramidal neurons, the basal dendrites have ≈ .
6% shorter total length(3513 ± µ m) in comparison to apical dendrites (5295 ± µ m) ( F , = 73 . P < . ≈ .
0% thinner (0 . ± . µ m) compared to apical dendrites(0 . ± . µ m) ( F , = 30 . P < . ≈ .
6% lower total volume ofbasal dendrites (5881 ± µ m ) as opposed to apical dendrites (11231 ± µ m ) ( F , = 5 . P = 0 . Figure 1: Layered structure of mouse neocortex constructed in silico with digital reconstructions of Layer 2-3 pyramidal neurons(NMO 51117, NMO 51116), Layer 5 pyramidal neurons (NMO 09483, NMO 09485, NMO 09480, NMO 09494) and Layer 6pyramidal neurons (NMO 85158, NMO 85162). Basal dendrites are rendered in red, apical dendrites in purple, and axons inblue. Neuron identification numbers are given from left to right of the rendered reconstructions.
Axons are specialized to deliver electric output to distant targets. The mean radius of axons (0 . ± . µ m) is ≈
25% thinner compared to the mean radius of dendrites (0 . ± . µ m) (paired t -test, t , = 10 . P < . n = 370) or complete axonal arborizations in whole brain reconstructions ( n = 158).The rationale for this analysis is that slicing does not affect the radii of neuronal projections. However,because the axonal arborizations are trimmed in slice sections, for the evaluation of total axonal length andtotal axonal volume, we have used only the subset of 158 automated whole brain reconstructions in mouse(Table 3). Axons of mouse projection neurons have ≈ . × greater total length (79020 ± µ m) and ≈ . × greater total volume (137351 ± µ m ) in comparison with dendrite total length (7319 ± µ m)(paired t -test, t , = 11 . P < . ± µ m ) (paired t -test, t , = 8 . P < . Approximately 20% of resting oxygen consumption (i.e. in the absence of heavy physical work by skeletalmuscles) is absorbed by the human brain (Laughlin et al., 1998). Brain activity is fueled almost exclusivelyby glucose (Magistretti & Allaman, 2015). Oxidative metabolism in mitochondria of 1 glucose molecule leadsto the production of 32 ATP molecules (Mergenthaler et al., 2013), each of which releases 0 .
123 basal dendrites apical dendrites
Leng t h ( × μ m ) a *** R ad i u s ( μ m ) b *** Leng t h ( × μ m ) d ***
012 axon dendrites R ad i u s ( μ m ) e *** V o l u m e ( × μ m ) c * V o l u m e ( × μ m ) f *** Figure 2: Paired box plots for morphometric measures in cortical pyramidal neurons. Comparison of total length L , averageradius r , and total volume V = πr L was performed for basal dendrites versus apical dendrites (a-c) or axon versus dendrites(d-f). Individual measurements are represented with black dots. Paired measurements performed in the same cell are connectedwith thin blue lines. The bottom and the top of each box represent the lower ( Q
1) and upper ( Q
3) quartile, whereas the blackline in the middle of the box represents the median. The interquartile range (IQR = Q − Q
1) contains the middle 50 % ofthe data, the whiskers extending from the minimum Q − . × IQR to the maximum Q . × IQR value indicate the spreadof the data, and the outliers are represented by data points that are located outside the whiskers of the box plot. Statisticalsignificance was estimated by repeated-measures analysis of variance (rANOVA): *, p < .
05; ***, p < . neuronal activities from glucose is only 1235 kJ/mol, even though combustion of glucose in oxygen releases2801 kJ/mol. From the speeds of glucose consumption (Herculano-Houzel, 2011) by the cerebral cortex ofdifferent species (Table 4), it can be estimated that the power of the mouse cortex is 0 .
004 W, rat cortexis 0 .
015 W, and human cortex is 4 .
427 W. These cortical values comprise approximately half of the powerof the whole brain (Table 5), namely, the power of the mouse brain is 0 .
008 W, rat brain is 0 .
025 W, andhuman brain is 9 .
628 W. This modest consumption of energy points to highly efficient energy utilization,and miniaturization of the brain’s logical circuitry.
Pyramidal neurons input, process and output cognitive information with the use of electric spikes. Thereare five main physiological processes that support each spike (Figure 3a):(1) Each neuron needs multiple excitatory dendritic inputs, which activate post-synaptic neurotransmit-ter receptors and generate excitatory post-synaptic potentials (EPSPs).4 able 1: Morphometric measures for basal dendrites of pyramidal neurons.
Species Brain region Neuron type µ m) Total volume( µ m ) Mean radius( µ m)Mouse Neocortex Layer 2-3 15 3398 ± ± . ± . ± ± . ± . ± ± . ± . ±
755 3568 ± . ± . ± ± . ± . ± ± . ± . ± ± . ± . ±
445 531 ±
290 0 . ± . ± ± . ± . ± ± . ± . ± ± . ± . Table 2: Morphometric measures for apical dendrites of pyramidal neurons.
Species Brain region Neuron type µ m) Total volume( µ m ) Mean radius( µ m)Mouse Neocortex Layer 2-3 15 3470 ± ± . ± . ± ± . ± . ± ± . ± . ± ± . ± . ± ± . ± . ± ± . ± . ± ± . ± . ± ±
786 0 . ± . ± ± . ± . ± ± . ± . ± ± . ± . Table 3: Morphometric measures for axons of pyramidal neurons.
Species Brain region Neuron type µ m) Total volume( µ m ) Mean radius( µ m)Mouse Neocortex Layer 2-3 15 65749 ± ± . ± . ± ± . ± . ± ± . ± . . ± ± . ± . able 4: Energy consumption by neurons in the cerebral cortex. Species Cortical mass(gray + whitematter) (g) Glucose useper gramper minute( µ mol/g · min) Total number ofcortical neurons( × ) Energy use perneuron (pW) Total energy useby cortex (W)Mouse 0 .
173 1 .
10 1 .
369 286 .
13 0 . .
769 0 .
95 3 .
102 484 .
77 0 . .
52 0 .
34 1634 270 .
91 4 . Table 5: Energy consumption by neurons in the brain.
Species Brain mass (g) Glucose useper gramper minute( µ mol/g · min) Total number ofbrain neurons( × ) Energy use perneuron (pW) Total energy useby brain (W)Mouse 0 .
416 0 .
89 7 .
089 107 .
50 0 . .
802 0 .
68 20 .
013 126 .
03 0 . .
91 0 .
31 8606 111 .
88 9 . + , K + or Ca ions.The energy expenditure in vivo varies from neuron to neuron depending on the exact morphometricmeasures and physiological activities. However, it is possible to estimate the energy budget for an averagecortical pyramidal neuron under the assumption that all of the energy released from glucose consumed bythe brain cortex is used to fuel electric spiking with the underlying biomolecular processes. Because themaintenance of resting membrane potential by neurons, and the accompanying glial support could be viewedas continuous processes interspersed by discrete action potentials, the general outline of the calculation ofthe energy budget per spike is as follows: Firstly, from the total energy budget of the cerebral cortex, theenergy needed for neuronal resting membrane potential and glial support is subtracted. Secondly, fromthe remaining energy, the maximal average frequency of firing electric spikes is computed. Thirdly, themaximal average firing frequency provides an estimate of the duration of the average interspike interval andthe corresponding energy expenditure for the neuronal resting membrane potential and glial support persingle spike. Finally, the data so obtained will be integrated for calculating the total energy expenditure tosupport a single spike together with its preceding interspike interval. Physiological electric activities are due to passage of metal ions across the plasma membrane. The restingmembrane potential of neurons is approximately −
70 mV. The influx of Na + ions leads to depolarization,whereas the efflux of K + ions leads to hyperpolarization of the transmembrane voltage. Neurons use 3 . × ATP molecules per second in order to keep steady their resting membrane potential (Attwell & Laughlin,2001). The power consumed by 1 . × neurons in the human cerebral cortex (Herculano-Houzel, 2011)for their resting potential is 0 .
358 W, which accounts for ≈ .
1% of the total cortical power.
Glial cells also spend energy to sustain their resting membrane potential at about −
60 mV (McKhannet al., 1997). Glial cells, which are 3 . × more numerous than neurons in the cerebral cortex (Azevedo et al.,2009), consume 1 . × ATP molecules per glial cell each second (Attwell & Laughlin, 2001). For thehuman cerebral cortex, the energy consumption by glial cells is 0 .
406 W, which constitutes ≈ .
2% of thetotal cortical power. 6 ynapseneuron dendriticinputs spikeglial cell
Energy budget for physiological activities per spike a b
Action potentialExocytosisDendriticsynapticactivity RestingmembranepotentialGlialsupport45.7%10.0%27.0% 8.1%9.2%
Action potentialExocytosisDendritic synaptic activityResting membrane potentialGlial support 12.87 pJ7.60 pJ2.81 pJ2.59 pJ2.28 pJ c restingmembranepotential Figure 3: Physiological activities underlying the input, processing and output of cognitive information through electric spikesby pyramidal neurons. (a) To generate an electric spike, each neuron (1) needs multiple excitatory dendritic inputs, whichactivate post-synaptic neurotransmitter receptors. The excitatory post-synaptic potentials (EPSPs) then (2) summate at thesoma and trigger an action potential at the axonal hillock. The action potential propagates to pre-synaptic axonal buttonsthat (3) release neurotransmitter through exocytosis of synaptic vesicles. Excess neurotransmitter is (4) recycled by glial cells,which support the proper functioning of neurons. In between spikes, pyramidal neurons expend energy in order to (5) maintaintheir resting membrane potential. The energy budget in picojoules (pJ) for these five main physiological activities per spike istabulated in (b) and displayed as a pie chart with percentages of the total energy consumed in (c).
Neuronal dendrites are unmyelinated and leak-prone, but they need to be depolarized in their proximalpart that is adjacent to soma, to a level slightly above a threshold of −
54 mV (Pathak et al., 2016) in orderto trigger an action potential at the axonal hillock. To reduce leakage, and achieve efficient transport of theelectric spike to distant targets, the axons are myelinated with the exception of the nodes of Ranvier, whereupon electric stimulation, the membrane readily depolarizes to +40 mV due to opening of voltage-gatedNa + channels. The number of Na + ions entering into a cylindrical neurite segment is given by the capacitorcharge formula N = 2 πrLf ∆ V C m q e = Af ∆ V C m q e (1)where r is the radius, L is the length, A = 2 πrL is the surface area, f is the fraction of unmyelinatedactive membrane, ∆ V is the voltage change, C m = 1 µ F/cm is the specific membrane capacitance, and q e = 160 .
218 zC is the elementary electric charge.Dendrites are completely unmyelinated f = 1. They are depolarized by ∆ V = 50 mV during thebackpropagation of an action potential (Attwell & Laughlin, 2001). Direct substitution in Eq. (1) establishesthat, for each action potential, in the basal dendrites with mean total length L = 3513 µ m and averageradius r = 0 . µ m enter N basal = 3 . × Na + ions, whereas in apical dendrites with mean total length L = 5295 µ m and average radius r = 0 . µ m enter N apical = 6 . × Na + ions.Axons are heavily myelinated with f = 0 .
018 as estimated from the mean length of unmyelinated nodes7f Ranvier, which is 1 . µ m, and the internode mean distance of 81 . µ m (Arancibia-C´arcamo et al., 2017).During an action potential, axons are depolarized by ∆ V = 110 mV (Schwindt et al., 1997). Again, directsubstitution in Eq. (1) establishes that, for axonal trees with mean total length L = 79020 µ m and averageradius r = 0 . µ m, enter N axon = 2 . × Na + ions per action potential.For a pyramidal soma with surface area A = 2970 ± µ m (Zhu, 2000), f = 1 and ∆ V = 110 mV,direct substitution in Eq. (1) establishes that there is an additional load of N soma = 2 . × Na + ions.For a realistic estimate of the total Na + entry into a pyramidal neuron per action potential, the amountof Na + ions computed from the capacitor charge formula should be multiplied by an overlap factor f overlap in order to account for simultaneous activation of Na + and K + channels with Hodgkin–Huxley kinetics N HH = f overlap × ( N basal + N apical + N axon + N soma ) (2)Computational simulations by Attwell & Laughlin (2001) have found that f overlap = 4. The total sodiumload in dendrites, soma and axon obtained from Eq. (2) amounts to N HH = 6 . × Na + ions. TheseNa + ions need to be pumped out of the neuron by protein Na + /K + -ATPase, which exports 3 Na + ionsand imports 2 K + ions for every ATP molecule that is consumed (Sengupta et al., 2013). Thus, for eachelectric spike, to remove the load of Na + ions each neuron needs N HH / . × ATP molecules, whichamounts to 12.87 pJ of energy (Figure 3b).
Exocytosis with subsequent endocytosis of a single synaptic vesicle consumes 1 . × ATP molecules(Attwell & Laughlin, 2001). Recycling of 4000 glutamate neurotransmitter molecules released per vesicle,through glutamate uptake by glial cells, glial conversion to glutamine, export of glutamine to neurons,neuronal conversion to glutamate and re-packaging into synaptic vesicles, consumes another 1 . × ATPmolecules (Attwell & Laughlin, 2001). Thus, the total energy consumption by a single synaptic vesicle is2 . × ATP molecules.Axons of cortical pyramidal neurons form between 7000 and 8000 synapses onto target neurons (Braiten-berg & Sch¨uz, 1998). The number of released synaptic vesicles N released is proportional to the total numberof axonal synapses N synapses and the probability of release p release of a synaptic vesicle per action potentialper synapse N released = p release × N synapses (3)Considering that the release probability of a synaptic vesicle is only 0 .
25 per action potential per synapse(Georgiev & Glazebrook, 2018), each pyramidal neuron will release on average 1875 vesicles (for projectionneurons most of the targets can be extracortical). Thus, for each electric spike, the exocytosis of synapticvesicles consumes 4 . × ATP molecules, which amounts to 2.81 pJ of energy (Figure 3b).
Action potentials cannot be spontaneously generated by healthy pyramidal neurons from their restingstate. Instead, a significant amount of preceding dendritic activity would be required to excite the neuron.An important factor in many related scenarios is the nature and effect of spatial compartmentalization.For instance, Polsky et al. (2004) observed that rat neocortical pyramidal neurons initially process theirsynaptic inputs within thin dendritic subunits regulated by a nonlinear sigmoidal-type threshold, and thenat a second stage they are linearly combined to deliver the overall neuronal response.Each excitatory synaptic input delivered at a dendrite spine-head depolarizes the soma by only 0 .
12 mV(Kubota et al., 2015). Thus, for a potential rise of 16 mV, from the resting membrane potential of −
70 mVto the spike threshold of −
54 mV, summation of at least 134 dendritic spine inputs would be needed.Indeed, based on detailed experimental electrophysiological data in vivo using two-photon activation of anintracellular caged NMDA receptor antagonist, it was confirmed that the dendrites of pyramidal neurons needto receive an excess of excitatory synaptic inputs N excess = 140 (activating NMDA and AMPA receptors) inorder to trigger an action potential (Palmer et al., 2014).In the awake state characterized by γ -frequency electric oscillations, the activation of powerful peri-somatic inhibition by fast-spiking interneurons (Georgiev et al., 2014; Hu et al., 2014), however, causes8yperpolarization or shunting that suppresses the effects of excitatory synaptic activation in pyramidal neu-rons. To take into account the effect of cortical inhibition in the presence of non-zero excitatory to inhibitory(E/I) ratio, the number of dendritic synaptic inputs N inputs can be modeled as N inputs = N excess × (cid:18) (cid:19) (4)The E/I ratio at the soma of layer 2/3 pyramidal neurons is 0 .
8, whereas at the soma of layer 5 pyramidalneurons it is 0 . . × Na + ions and 10 Ca ions enter into the dendrite(Attwell & Laughlin, 2001). Calcium signaling in dendrites leads to Ca load that needs to be removed byNa + /Ca exchanger, which exports 1 Ca ion and imports 3 Na + ions. The 3 Na + ions are subsequentlyexported by Na + /K + -ATPase consuming 1 ATP molecule. For 578 synaptic inputs per action potential,the extrusion of the Na + and Ca ion load requires 7 . × ATP molecules. Recycling of the vesicles(discussed in the preceding subsection) further requires 1 . × ATP molecules. The energy expenditurefor Ca signaling during backpropagation of the action potential from axonal hillock to dendrites addsanother 2 . × ATP molecules (Attwell & Laughlin, 2001). Thus, for each electric spike, dendriticsignaling consumes 1 . × ATP molecules in total, which amounts to 7.60 pJ of energy (Figure 3b).
Taking stock of matters so far, we see that neural information is indeed costly. The energy expenses bya single neuron to support the dendritic synaptic activity required to elicit an action potential, to sustainthe propagation of the action potential towards pre-synaptic axonal buttons, and to execute the associatedrelease of synaptic vesicles for neurotransmitter signaling, sum up to 3 . × ATP molecules, whichrelease 23 .
28 pJ of free energy. For 1 . × neurons in the human cerebral cortex (Table 4), therequired energy to fire once is 0 .
38 J. After subtraction from the total cortical budget of the energies spenton the resting membrane potential by neurons and glial cells, there is a remaining energy power of 3 .
663 Wthat can be spent by the cerebral cortex on firing action potentials with an average frequency of 9 . ≈ .
3% of the maximal firing frequency of 67 Hz that can be attained by layer 5 pyramidalneurons (Schwindt et al., 1997). For average spiking frequency of 9 . ≈
104 ms. Thus, for each electric spike, the maintenance of neuronal resting membrane potentialin the preceding interspike interval uses 2.28 pJ and the glial support uses 2.59 pJ of energy (Figure 3b).The total energy budget for a single electric spike together with the preceding interspike interval amountsto 28.15 pJ. In summary, 45.7 % of the energy budget is dedicated for propagation of the action potential,27.0 % for support of dendritic synaptic activity, 10.0 % for exocytosis of synaptic vesicles, 9.2 % for glialsupport, and 8.1 % for maintenance of the neuronal resting membrane potential in the interspike interval(Figure 3c).Original estimates by Attwell & Laughlin (2001) pointed to 3 . × ATP molecules (210.85 pJ) con-sumed by a neuron with a mean firing rate of 4 Hz. The energy budget stipulated 47% for the productionof action potentials, 34% for the activity of dendritic post-synaptic receptors, 6% for presynaptic exocytosisincluding recycling of excess glutamate, and 13% for maintenance of the resting state of neurons and sup-porting glial cells. The main difference between that previous study and our present results stems from theprecise morphometric data that we have used resulting in higher average spiking frequency by pyramidalneurons due to lower energy needs to support action potentials.
Moving a single elementary electric charge (electron, proton or monovalent ion) across the plasma mem-brane through a potential difference of 110 mV dissipates 0 .
11 eV of energy. The energy of 23 .
28 pJ consumed9er action potential is sufficient for the motion of 1 . × elementary electric charges. The transportof an elementary electric charge across the plasma membrane, however, may not be the elementary bit ofneuronal logical operation. Landauer’s limit asserts that the minimum possible amount of energy requiredby thermodynamics to erase one bit of information (e.g. through application of an irreversible gate such asAND gate or OR gate) is E min = k B T ln 2 (5)recalling that k B = 1 . × − J/K is Boltzmann’s constant and T is the absolute temperature (Lan-dauer, 1961). Otherwise expressed, if ∆ E env denotes energy dissipated into the environment, and ∆ S sys thethermodynamic entropy equivalent to information erased from the system memory, then∆ E env ≥ T ∆ S sys (6)The total number of erased bits of information I erased from the system is bounded by (Bormashenko, 2019;Street, 2020) I erased ≤ ∆ E env k B T ln 2 (7)At physiological temperature of 310 K, Landauer’s limit is 2 .
968 zJ (18 .
526 meV). Therefore, the energyof 23 .
28 pJ consumed per action potential is sufficient for the execution of 7 . × Landauer elementarylogical operations. Noteworthy, the passage of a single elementary electric charge across the plasma mem-brane is equivalent to ≈ .
94 such elementary logical operations. Since each S4 protein α -helix voltage-sensorin voltage-gated ion channels (Figure 4) usually contains 6 positively charged amino acid residues (Catterall,1988), the proton tunneling between neighboring positively charged sites in the S4 voltage-sensor (Kariev& Green, 2012, 2018, 2019; Kariev et al., 2007) is ideally suited to represent a single Landauer elementarylogical operation in cortical neural networks. Protons interact with water and biological matter, mainly in anon-classical manner including exchange-correlation effects, chemical bonding in hydronium-like complexes,and tunneling (Lobaugh & Voth, 1996). Once the S4 protein α -helix voltage-sensors adopt an open channelconformation, the subsequent flow of metal ions across the ion channel leads to amplification of individualevents of proton quantum tunneling that occurred in the S4 voltage sensors (Kariev & Green, 2012, 2018,2019; Kariev et al., 2007). This is how nanoscale quantum events may be amplified to exert macroscopiceffect on neuronal behavior and brain function (Georgiev, 2013, 2020). If the energy power of 3 .
663 Wavailable for electric spiking is completely miniaturized at the Landauer limit of 2 .
968 zJ, the human braincortex will be able to execute the equivalent of over 1.2 zetta elementary logical operations per second.
3. Discussion
In this work, we have evaluated the cognitive computational capacity of the brain based on its experi-mentally measured glucose consumption (Herculano-Houzel, 2011). The tightness of the bound is justifiedby two biomedical facts. Firstly, the brain does not have an internal store of glucose, but needs to rely onblood glucose level maintained by physiologically regulated release from the liver glycogen depot (Guyton &Hall, 2006). Secondly, the brain does not have anything like a long-term energy battery capable of support-ing cognitive computation in the absence of glucose, because loss of clinical consciousness (syncope) occurswithin seconds of a sudden drop in blood glucose levels, or a brief cessation of cerebral blood flow (Kapoor,2000). This implies that the rate of glucose consumption indeed puts a tight upper bound on the brain’scapacity for cognitive computation.In the awake state, the energy power of 4 .
427 W consumed by the human cerebral cortex permits amaximal average spiking rate of 9 . .
87 Hz by neurons in the visual cortex of awake rhesus monkeys (Chen et al., 2009), albeit itis somewhat higher than the average spontaneous firing rates of 3 .
28 Hz in rat (Aasebø et al., 2017) or1 .
88 Hz in mouse (Durand et al., 2016). Evoked activity of neurons in primary visual cortex of awakemice in response to an optimal drifting grating, however, exhibited average firing rates that were stronglydependent on locomotion: 2 . . a + extracellular spaceintracellular spaceN +++ P +++ +++ C +++ Cav
I II III IV α− subunitP P P Nav Kv
I IIIIIIV plasma membrane αα αα
I IIIIIIV
Cav
Nav Kv α− subunitN +++ P N CP +++ +++ C +++ I II III IVP P P +++ α− subunit Na+Na+ Ca + K+K+ Figure 4: Electric activities of pyramidal neurons are generated by sodium (Nav), potassium (Kv) and calcium (Cav) voltage-gated ion channels incorporated into the plasma membrane, which consists of a phospholipid bilayer with thickness of 10 nanome-ters. Structurally, individual voltage-gated ion channels contain four protein domains I-IV. Each domain has six transmembrane α -helices (1-6). The pore of the ion channel is configured by protein loops (P) connecting the 5th and 6th α -helices. Voltagesensing is accomplished by the 4th α -helix (S4), which usually carries six positively charged lysine or arginine amino acidresidues. Each proton tunneling event between neighboring positively charged sites in the S4 voltage-sensor is ideally suited torepresent a single Landauer elementary logical operation. spherical treadmill with their heads fixed (Niell & Stryker, 2010). Thus, the estimated maximal averagespiking rate of 9 . . × elementary logical operations. The cerebral cortex appears to have attainedthe maximal computational efficiency allowed by Landauer’s thermodynamic limit: quantum tunneling ofa proton between neighboring positively charged S4 sensor sites in voltage-gated ion channels constitutes asingle Landauer elementary logical operation, whereas the transport of a monovalent metal ion through theopen ion channel pore constitutes six such operations.11andauer’s limit sets ultimate energy constrains on the functioning of physical computing devices in thepresence of a thermal bath (Gaudenzi et al., 2018; Hong et al., 2016; Lent et al., 2019; Sagawa & Ueda,2008, 2010). The original formulation put forward by Landauer (1961) is motivated by consideration ofthe finite capacity of working memory of computing devices, namely the act of resetting of the workingmemory to its initial empty state requires compression of the phase space of the memory device, whichwill decrease its entropy. The second law of thermodynamics, however, requires that the total entropyof the memory device and its environment increases in time (Leff & Rex, 2002). Therefore, resetting theworking memory must be accompanied by a corresponding entropy increase in the environment, in theform of heat dissipation, which is at least k B T ln 2 joules per bit (Landauer, 1961). The brain cortex,which is responsible for the stream of consciousness, allows us to store cognitive information only for shorttime periods before we forget it, or replace it with new sensory information. Thus, one interpretation ofLandauer’s principle is that the working of the human mind spends energy to forget (Plenio & Vitelli, 2001),where the energy dissipation occurs in the act of irreversible resetting of the cortical working memory.An alternative formulation based on the theory of dissipative quantum channels, however, establishes thatcommunication of classical information across a noisy quantum channel (Jagadish & Petruccione, 2018) thatis immersed in a heat bath with effective temperature T , also requires energy expenditure of at least k B T ln 2joules per bit (Levitin, 1998; Porod et al., 1984). Otherwise, the signal transmitted from the sender gateto the receiver gate could not be distinguished from the ambient thermal noise (Levitin, 1998; Porod et al.,1984). Thus, another interpretation of Landauer’s principle is that the working of the human mind spendsenergy to transmit information between different noisy neuronal compartments (dendrites, soma, axon) orto communicate unambiguously with effector organs (e.g. muscles through intermediate extracortical centerssuch as α -motor neurons in the spinal cord). It is likely that, in the course of an electric spike, corticalpyramidal neurons spend energy both for resetting their S4 voltage sensors in the resting ion channel stateand for transmission of the electric signal from dendrites toward the axon terminals. Much related areimplications of this neurophysiology together with Landauer’s principle for human cognition, as discussedin Collel & Fauquet (2015); Street (2016, 2020) with significant pointers towards variational free energy andits role in perceptual Bayesian inference (Friston, 2010, 2013).Because physical dynamics at the nanoscale is able to manifest characteristic quantum mechanical ef-fects, our results provide a rigorous foundation, as far as energy considerations are concerned, for futuredevelopment of quantum models of the transmembrane electromagnetic field and its interaction with mobileelectric charges inside protein voltage-gated ion channels (Georgiev, 2017; Kariev & Green, 2012, 2018, 2019;Kariev et al., 2007) or membrane-bound SNARE proteins whose zipping mechanism triggers neurotrans-mitter release (Georgiev & Glazebrook, 2018, 2019a,b). Technological advances in available supercomputershave already led to routine simulation of quantum dynamics of small biomolecules in electrolyte solutionwith the use of quantum chemistry software implementing density functional theory (Kolev et al., 2013,2018, 2011). Applications of recent theorems in quantum information, as based on generalized uncertaintyrelations (Carmi & Cohen, 2019) to quantum brain states (Georgiev, 2013, 2020) may further shed light onthe perplexing open problems in the cognitive sciences.To summarize, we have implemented fundamental physical principles, including the thermodynamicallyallowable Landauer’s limit of energy spent on elementary logical operations, to show that not all biomolecularprocesses may contribute to cognitive computation, but mainly those involving transmembrane proteins,such as voltage-gated or ligand-gated ion channels, integrated into the electrically excitable neuronal plasmamembrane. Even though the human cerebral cortex may perform over 1.2 zetta logical operations persecond, exceeding over four orders of magnitude the capacity of modern supercomputers, we expect theimplementation of large-scale and ultra in-depth brain simulations to significantly advance in the foreseeablefuture.
4. Conflict of Interest
The authors certify that they have no affiliations with or involvement in any organization or entity withany financial interest, or non-financial interest in the subject matter or materials discussed in this work.12 . Methods and Materials
Morphometric parameters such as radii and lengths of neuronal projections constrain the electric perfor-mance of neurons and determine the number of physical charges that need to cross the plasma membranein order to elicit a certain change in the transmembrane voltage. For accurate assessment of the aver-age radii and total lengths of different neurites (basal dendrites, apical dendrites, and axons), we haveanalyzed the full collection of pyramidal neuronal reconstructions in rodent (mouse, rat) or human braincortex from NeuroMorpho.org 7.8 digital archive (Ascoli et al., 2007) that pass the following selection cri-teria: Firstly, we have selected only control experimental conditions with animals that did not expressgenetically-engineered disease-related protein mutations and were not exposed to pharmacological agentsor harmful stimuli (e.g. stress). Secondly, only animals whose age corresponds to human age of over 3months old were included. The utilized piecewise linear conversion formulas into corresponding humanage are given for mice by Sengupta (2013), and for rats by Dutta & Sengupta (2016). Thirdly, to en-sure minimal trimming of dendritic trees for analysis of apical and basal dendrites, we have included onlyreconstructions with minimal slice thickness of 300 µ m. Analysis of complete axonal arborizations wasperformed in neuronal reconstructions from brain-wide imaging data (Economo et al., 2016; Gerfen et al.,2018). To verify the quality of all reconstructions, neurons were visualized in Neuromantic version 1.6.3( ) and .swc files with non-standard labelingof neurites or visually incomplete dendritic tree (e.g. apical dendrite was trimmed near its base) wereexcluded from further analysis. Standardized .swc files are tables with 7 columns of numerical data forcable-like cylindrical segments that comprise the neuronal reconstruction (Table 6). The lengths and vol-umes of neurite segments was quantified with the use of custom Excel macros fetching the cable radii andcomputing the Euclidean distances from the x, y, z coordinates given in the .swc files. Morphometric dataare reported as mean ± standard deviation. Table 6: Table structure of standardized .swc files. x position y position z position radius r parent segmentintegervaluestartingfrom 1 1 - soma2 - axon3 - basal dendrite4 - apical dendrite coordinatein µ m coordinatein µ m coordinatein µ m segmentradius in µ m parent segmentnumber; − Digital reconstructions of pyramidal neurons in control experimental conditions were selected from threeanimal species: mouse (252 neurons), rat (491 neurons) and human (6 neurons). This dataset of 749neurons includes contributions from 32 labs: Amaral (Ishizuka et al., 1995), Arnold Johnston (Arnold et al.,2019), Barrionuevo (Henze et al., 1996), Blackman (Blackman et al., 2014), Buchs (Larkum et al., 2004),Chandrashekar (Economo et al., 2016), Claiborne (Carnevale et al., 1997), De Koninck (Bories et al., 2013),Dendritica (Vetter et al., 2001), Feldmeyer (Marx & Feldmeyer, 2012; Marx et al., 2015), Groen (Groenet al., 2014), Hay (Hay et al., 2013), Helmstaedter (Helmstaedter et al., 2008), Hoffman (Hoffmann et al.,2015), Jaffe (Chitwood et al., 1999), Johnston (Dougherty et al., 2012; Malik et al., 2016), Kawaguchi (Hiraiet al., 2012; Ueta et al., 2013), Kole (Hallermann et al., 2012; Hamada et al., 2016; Hamada & Kole, 2015;Kole, 2011; Kole et al., 2007, 2004), Korngreen (Bar-Yehuda & Korngreen, 2008), Krieger (Groh et al., 2009;Krieger et al., 2007), Luo (Gong et al., 2016), Markram (Anastassiou et al., 2015), Martina (Kelly et al.,2016), MouseLight (Gerfen et al., 2018), Orion (Santamar´ıa-Pang et al., 2015), Segev (Eyal et al., 2016),Soltesz (Lee et al., 2014), Spruston (Golding et al., 2005), Staiger (Staiger et al., 2016), Storm (H¨onigspergeret al., 2015), Topolnik (Francavilla et al., 2018; Tyan et al., 2014) and Urban (Tripathy et al., 2015; Zhouet al., 2015). 13 .3. Modeling of cortical layers
Vector .svg images of individual neurons were rendered with HBP Neuron Morphology Viewer (Bakkeret al., 2017; Bakker & Tiesinga, 2016) and scaling information was extracted with NeuroM, a Python-basedtoolkit for the analysis and processing of neuron morphologies developed by the Blue Brain Project ( https://neurom.readthedocs.io/en/stable/ ). Modeling of the brain cortex in mouse was then performed inAdobe Illustrator based on measured thickness of cortical layers in Nissl stained coronal slices (Franklin &Paxinos, 2007; Georgiev et al., 2016). All data from NeuroMorpho.Org digital archive was used in compliancewith the online Terms of Use ( http://neuromorpho.org/useterm.jsp ). In particular, all original papersthat describe the reconstructions are cited, the complete name of the digital archive is clearly stated,attribution to the developers of the archive is given (Ascoli et al., 2007), and specific reconstructions arereferenced with their NeuroMorpho.Org ID numbers.
Statistical analysis of neuronal morphology was performed using SPSS ver. 23 (IBM Corporation, NewYork, USA). Comparison of morphometric measures for apical and basal dendrites was performed withrepeated-measures analysis of variance (rANOVA) implemented as a general linear model in which within-subject variable was dendrite type, between-subject factors were animal species, brain region and neuronaltype, and covariate was slice thickness. Comparison of axons with dendrites was performed with paired t -tests for a subset of the neuronal reconstructions for which the axonal trees were complete. Paired boxplots were created with the use of ggpubr library in R ver. 4.0.2 (R Foundation for Statistical Computing,Vienna, Austria, ). The energy consumption by pyramidal neurons in different animal species (mouse, rat or humans) wasestimated based on brain mass, glucose use per gram per minute, and total number of neurons in the brain orthe cerebral cortex reported in Herculano-Houzel (2011). For each molecule of glucose, oxidative metabolismin mitochondria produces 32 ATP molecules (Mergenthaler et al., 2013). Hydrolysis of 1 ATP moleculereleases 0 . .
218 zJ. Brain power wasreported in watts (W), where 1 W is defined to be the energy transfer at a rate of 1 J per second. For allreported quantities standard SI prefixes were used.
Acknowledgements
E.C. acknowledges support from the Israel Innovation Authority under project 70002 and from theQuantum Science and Technology Program of the Israeli Council of Higher Education.
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