Limited synapse overproduction can speed development but sometimes with long-term energy and discrimination penalties
[['The phenomenon of net synapse overproduction in brain development is well-documented [1–4] but , as far as we can find , almost nothing definite is said about its purpose .', 'Nevertheless , misregulation of this overproduction phenomenon is postulated to be a possible cause of autism [5–7] .', 'The research reported here provides a functional perspective for two opposing goals in the regulation of synapse number .', 'The first goal is that synapse overproduction speeds the time to develop a quasi-stable connectivity ( from here on referred to as stable connectivity or just stability; see Methods for the need to qualify stable connectivity as quasi-stable rather than an absolute stability ) .', 'The second goal is to avoid too many synapses per neuron for reasons of', '( i ) energy efficiency of neuron-use [8 , 9] ,', '( ii ) per-category allocation of neuronal responses , and', '( iii ) specificity of neuronal responses [10] .', 'Empirical studies suggest that following overproduction , net synapse loss is integral to normal development [11 , 12] including development of executive function [12] and memory performance [13] .', 'Indeed , the results here support , in a quantitative fashion , Purves’s and Lichtman’s qualitative suggestion that “subsequent elimination of some innervation is a strategy well-suited to ensuring both the prompt innervation of all the cells in a target and ultimately , a quantitatively appropriate distribution of synapses among the target cells” [p . 135 , 14; underscoring is our’s] .', 'Moving from this qualitative statement toward the quantitative , in general , it might seem that faster development is better for the parents ( particularly the dam ) , but there are other considerations; that is , the performance of neurons in the adult will also constrain development .', 'It seems that synapse overproduction appears in the computational literature only twice , and both times it is found in a figure without comment [10 , 15] .', 'The later study [10] shows that the synapse development paradigm of adaptive synaptogenesis [16 , 17] can produce neuronal allocation in proportion to the input category frequency; this paradigm creates such a neuronal allocation in an unsupervised manner , without inhibition , and starting from a connectionless set of neurons .', 'Here , we continue to investigate adaptive synaptogenesis .', 'This model is inspired by BCM theory [18] and its functional control of activity ( and we are not alone in considering functional controls over structural plasticity [19–21] ) .', 'As used here , adaptive synaptogenesis incorporates', '( i ) Hebbian synaptic modification ,', '( ii ) stochastic , postsynaptic activity-dependent synaptogenesis , and', '( iii ) anti-Hebbian driven synapse elimination [16 , 22] .', 'Previous studies have shown that this paradigm can produce what appears to be a homeostasis in synapse number and neuron firing while encoding the statistics of the input environment into its connections [10 , 23–26] .', 'The paradigm can also reproduce the results of postnatal ocular dominance experiments [27] , and it achieves compressive coding with small information losses [24] .', 'The present study begins with variants of an earlier result concerning the use of adaptive synaptogenesis in discrimination learning and the resulting neuronal allocation that reflects input statistics [10] .', '( Indeed , all the quantified results reported here were qualitatively observed in [10] for A , B1 , B2 , and B3 datasets; see Fig 6 in [10] . )', 'In the context of allocation that is proportional to input statistics and low error rates as outcomes of development , we evaluate rates of synapse growth and modification ( i . e . , γ and ε ) .', 'Higher rates of synapse growth and plasticity reduce the time to reach stable connectivity , but these higher rates raise the energy-costs of using the stabilized network and downgrade proportional neuronal allocation and specificity of neuronal responses .', 'The same result is obtained by manipulating the initial number of synapses .'], ['The adaptive synaptogenesis model was developed to explain the results of postnatal manipulations of sensory experiences , for example the effect of monocular deprivation on neuron allocation in V1 neocortex [27] .', 'Our interest here is in extending this theory to earlier times in development .', 'A standard result in experimental neuroscience is that neurons are allocated proportionately to the frequency of input patterns [28] .', 'Studies of adaptive synaptogenesis produce this proportionality [10 , 26 , 27] .', 'That is , there is greater neuronal representation for categories that occur more frequently .', 'Before studying the simpler and more easily interpreted dataset A1 , we remind the reader of the biological relevance of the theory by demonstrating biased allocation of orientation sensitive neurons via simulations of prenatal and postnatal development .', 'The simulations make use of three environments:', '( i ) a prenatal , retinal wave-generated environment with no bias in orientation ( dataset C3 ) ,', '( ii ) a normal postnatal environment with slight biases in the vertical and horizontal orientations ( dataset C4 ) , and', '( iii ) a postnatal , experimentally manipulated , stripe-tilt environment with biases of 45 ( dataset C1 ) and 135 ( dataset C2 ) degrees .', 'The three set of patterns were presented in sequence , one after another , with the algorithm running throughout; within each set , the patterns were randomly permuted .', 'See Methods for more detail about the datasets and the references to the empirical motivations .', 'All neurons begin with one connection .', 'By the time the connections stabilize in the prenatal environment , a neuron has , on average , 9 . 4 connections , while in the normal postnatal and experimentally manipulated postnatal environments , a neuron has , on average , 11 . 1 and 12 . 1 connections .', 'Stabilization in each environment produces different neuron allocations that reflect the orientation statistics of each environment ( Fig 1 ) .', 'In the prenatal , uniform environment , neuron allocation is evenly spread across all orientations ( Fig 1A ) .', 'In the normal postnatal environment , more neurons are allocated to horizontal and vertical orientations ( Fig 1B ) , and in the experimentally manipulated , stripe-tilt environment , orientations of 45 ( Fig 1C ) or 135 ( Fig 1D ) degrees receive the most neuronal allocation .', 'Because there is a certain complexity to the orientation datasets due to the overlaps between adjacent orientations [10] , we turn to a simpler , non-overlapping environment with five categories of patterns , each with a different frequency of presentation to the network .', 'The results of this section establish a baseline of neuronal allocations required of later results .', 'Within dataset A1 , neuronal allocation is a linear function of category frequency ( the fraction of presentations of patterns generated by a certain prototype ) .', 'Fig 2 illustrates this linear relationship .', 'The five orthogonal categories with fractional frequencies of 0 . 1 , 0 . 15 , 0 . 2 , 0 . 25 , and 0 . 3 receive fractional neuron allocations of 0 . 01 , 0 . 08 , 0 . 21 , 0 . 29 , and 0 . 41 , respectively .', 'The linear regression line of Fig 2 has a slope of 2 . 0 and a r2 value of 0 . 99 .', 'Raising the rates of synaptogenesis ( γ ) reduces the time to reach stable connectivity ( Fig 3 ) .', 'In conjunction with γ , the rate of synaptic modification ( ε ) also determines the time to reach stability .', 'Thus , in Fig 3 , the value of ε is the empirically observed optimum for each value of γ .', 'See Methods and S1 Appendix Fig 3 for details of the empirical optimization of ε values .', 'Fig 3 plots time to stable connectivity as a function of the inverse of γ .', 'There are two plots in Fig 3 because of the 500-fold range of γ values simulated; i . e . , Fig 3A shows the results for the smaller γ values , and Fig 3B shows the results for the larger γ values .', 'The two figures are in good agreement; the regression lines of Fig 3A and 3B have slopes of 0 . 06 and 0 . 05 , respectively , and the same r2 values of 0 . 99 .', 'Using much larger γ values ( e . g . 0 . 1 ) eliminates the linear relationship shown in Fig 2 .', 'Faster development mediated either by large γ is problematic when energy-costs are considered since faster development ( i . e . , shortest time-to-stability ) correlates with greater energy-costs ( Fig 4 ) .', 'Note that in Fig 4 , there are there different axes for three different costs , each of which is defined in Methods .', 'A postsynaptic neuron’s energy-costs arise from its computation and communication costs .', 'Computational costs of the neuron j arise from', '( i ) maintenance of its total number of synapses mj , and', '( ii ) the average use of these synapses , a value proportional to E[Yj] .', 'Communication costs arise from time-proportional leak and action potentials , where action potentials require about 100 times the energy-cost of leak per unit time [8] .', 'Thus , communication costs increases with firing-rate .', 'See Methods for further explanations of the measurements of energy-cost .', 'Fig 4 plots each of these relative costs as a function of γ; the greater the value of γ , the greater the firing-rates , the greater the average internal excitations , E[Yj] , and the greater the synapse numbers .', 'Combining these results ( Fig 4 ) with those of the previous section ( Fig 3 ) reveals an inverse relationship between time-to-stability and energy-costs of neuron-use ( Fig 5 ) .', 'As firing-rate increases for a neuron , its error-rate also increases .', 'An error-rate is defined as the fraction of the time a neuron fires to a pattern belonging to a non-preferred category ( see Methods ) .', 'The largest error-rate is six times the smallest .', 'The error-rates are steadily increasing with number of synapses , and this increase is what causes the gradual loss of specificity .', 'The error-rates are 0 . 0097 , 0 . 0135 , 0 . 0279 , 0 . 0435 , 0 . 0616 for the sequence of average number of synapses: 15 , 16 . 8 , 20 . 27 , 23 . 45 , and 25 . 8 .', 'With larger γ , the neuronal allocation that was a linear function of category probability ( see Fig 2 ) is lost .', 'For example , when γ is 0 . 1 , categories one through five have fractional neuronal allocations of 0 . 02 , 0 . 06 , 0 . 10 , 0 . 41 , and 0 . 40 , respectively .', 'In addition to increasing γ , initializing a simulation with more , random connections also reduces time-to-stability .', 'Initializing neurons with as many as 100 ( out of 1 , 000 possible ) randomly chosen synapses , as opposed to an initialization of just one synapse , speeds development while still achieving the appropriate neuron allocation ( as illustrated in Fig 2 ) .', 'In this particular case , the time-to-stability is cut by 60% .', 'Fixing values of γ and ε at 0 . 001 , SSNs of 1 , 50 , 100 , and 150 lead to an average time-to-stability of 90 . 9 , 55 . 4 , 36 . 8 , and 27 . 6 blocks , respectively .', 'The relationship between time-to-stability and energy-costs is consistent for both manipulations of γ ( continuous lines in Fig 5 ) and SSN ( dashed lines in Fig 5 ) .', 'A simulation with higher SSN has even shorter time-to-stability and even larger communication and computational costs than simulations with larger values of γ .', 'At an SSN of 100 and γ and ε values of 0 . 001 , a neuron reaches stability in 36 . 8 blocks on average , whereas a neuron at an SSN of 1 , a γ value of 0 . 001 , and an ε value of 0 . 0038 reaches stability in 56 . 7 blocks on average .', 'However , the energy-costs of the neuron developed with 100-SSN are much higher than the neuron developed at 1-SSN .', 'The percent increase for each cost from 1-SSN are 180% for firing-rate , 267% for excitation , and 258% for total weight per neuron , a substantial increase in each measure .', 'Though simulations with SSN values greater than 100 produce even faster development , the discrimination performance of neurons in these simulations is severely prone to error .', 'For example , with an SSN of 150 , the average firing-rate is 0 . 5 when stability is achieved; that is , a neuron at 150 SSN fires 50% of the time .', 'Each category in these simulations is , however , presented 30% of the time or less .', 'Thus , even when one assumes that such neuron fires 100% of the time to the most frequent category with its 30% appearance rate , it must be firing 20% of the time to other categories .', 'If the neuron prefers a category that is presented less than 30% of the time , the error rate can only be worse .', 'Thus , SSN-based overproduction also diminishes the specificity of neuron firing .', 'Overproduction of synapses is a well-known phenomenon of development and , at first glance , seems energetically wasteful .', 'This section points out why limited overproduction may exist .', 'For the adaptive synaptogenesis paradigm , synapse overproduction characterizes fast development; the greater the overproduction , the faster the development .', 'By definition , an overproduction occurs when a neuron gains more synapses than are present at stabilization; clearly at some point during development , there must be more shedding than synaptogenesis .', 'Fig 6 illustrates a single neuron example .', 'Starting with one connection , this neuron eventually stabilizes at 14 synapses .', 'However , it first achieves 14 synapses at block 52 and continues gaining synapses; it eventually reaches a maximum synapse number of 30 at block 101 .', 'Following this maximum , there is a period of oscillations , but on average , there is a decrement in synapse number until around block 200; at this point in development , shedding proceeds with very little synaptogenesis leading to a stable synapse number of 14 at block 235 .', '( The simulation continues for 1 , 088 more blocks with synaptic connections remaining unchanged . )', 'The size of this overproduction effect increases in concert with the speeding of development; this correlation is true for both γ or SSN manipulations ( Fig 7A ) .', 'Fig 7 presents direct comparison of the average maximum ( red/square line ) value to its corresponding average stable value ( black/circle line ) .', 'At a γ value of 0 . 001 and SSN of 1 , for example , on average , the maximum synapse number is 35 accompanied by an average stable value of 17; by calculation then , the average overproduction is 18 .', 'For this same SSN of 1 and the largest acceptable γ of 0 . 004 , the average overproduction is 37 . 7 .', 'For a γ value of 0 . 001 and SSN of 100 , the average overproduction is 75 . 6 synapses .'], ['Earlier studies of neuron optimization have concentrated on the energy-cost of information processing [29–32] and the energy-cost of communication [8] .', 'These sensible perspectives are not broad enough , at least for us , given the added context of synaptic development .', 'With this additional context , a lifespan perspective ( prenatal to adult ) motivates our questions about adaptive synaptogenesis [33–38] .', 'The demonstrations here illustrate our working hypothesis that natural selection creates a balance between speed of connectivity development and the later costs of using the relatively stabilized connectivity .', 'Inspired by this line of thinking , the report here notes the existence of an antagonistic interaction between two phenotypes that are both determinates of fitness in the modern Darwinian sense ( i . e . , lifespan and inter-generational theories of fitness; see above references and S3 Appendix for more about our research philosophy including our perspective on optimizations [39] ) .', 'The two phenotypes are', '( i ) the duration of neural development , a time period requiring the reproductive efforts of time and energy by the parents as they intensively nurture their developing offspring [37 , 38] , and', '( ii ) the implied adult behavioral performance and its adult costs associated with these same offspring .', 'In the case of', '( i ) , the costs include the energy-expense to the parents of providing for the entire organism of each of the offspring ( not just neural energy-costs ) and , additionally , the opportunity-loss incurred by the delay in starting a new brood; both of these costs increase with longer prenatal development-time and increase with the duration from birth to weaning .', 'In the case of', '( ii ) , the costs take the form of poorly apportioned neuronal allocation , assuming the importance of well-apportioned neural codes ( cf . Figs 1 and 2; see also discussion in 10 ) , and the energy associated with activating synapses ( cf . Figs 4 and 5 ) .', 'As far as we are aware , the antagonism between time-to-develop and later adult quality-of-performance has not yet influenced quantitative control of neural development .', 'The parametric study here , which hypothesizes such an antagonism , offers a rational explanation for limited synapse overproduction , at least for those who believe in a lifespan perspective for defining fitness .', 'Net synapse elimination characterizes certain phases of cortical development .', '( Net synapse elimination refers to the net loss in the number of synapses , as opposed to synapse elimination that is part of synapse turnover , a process that may not result in a change of synaptic totals . )', 'Both non-human studies [1 , 40–44] and human studies [45–47] show net synapse elimination occurs in various parts of the brain at various times in development .', 'For example , a human brain undergoes synapse growth prenatally and for a few months after birth .', 'Net synapse elimination then begins postnatally and accelerates at onset of puberty [48] .', 'Some studies observe synapse elimination even in young adults , both in humans [49] and rats [50] .', 'A primary hypothesis of the present research is that synapse overproduction reflects the speeding of development in order to reach an approximate steady-state of connectivity .', 'However , our model is not the only synaptogenesis model that finds the existence of overproduction .', 'Another postsynaptic activity-dependent paradigm also creates overproduction .', 'Although observed without comment , this other model [15] produces a similar correlation between overproduction and time-to-stability as elucidated here .', 'In focusing on synapse overproduction , the research here first points out that two fundamental parameters control overproduction—the {ε , γ} pair and SSN .', 'Second , it points out that too much overproduction , which does indeed continue to increase the speed of development , can cause later problems .', 'From this perspective , observing net synapse elimination in normal development is observing the result of a compromise , and from our perspective , an evolvable one .', 'The two manipulations ,', '( i ) {ε , γ} and', '( ii ) SSN , are natural controls of synaptic development; moreover , there is the requirement that {ε , γ} must be appropriately matched for best results , yet another part of the overall optimization problem studied here ( see Methods ) .', 'The similarity of the results using the two distinct manipulations , i . e .', '( i ) and', '( ii ) , is notable .', 'Both control mechanisms lead to the same problems when applied in an excessive manner .', 'Specifically , too much synapse overproduction leads to excessive adult energy-costs because', '( i ) acquiring many weak synapses at random implies that there will be more positively correlated input lines for whatever category wins the competition for a neuron’s allocation ( Fig 7 ) and because', '( ii ) more stabilized synapses consume more energy .', 'Moreover , accompanying the largest overproduction are two more problems: first , neuron allocation unduly favors the high probability categories while in the second case , a neuron is less likely to be category specific in its responses when it stabilizes with too many synapses .', 'Importantly , the SSN observations point to the generality of the results .', 'When SSN is large , the parameters controlling synaptogenesis—γ , ρ and Zj¯ , become irrelevant to the formation of synapses .', 'The mechanism of synaptogenesis is irrelevant because with large SSN , only the bidirectional Hebbian modifications and the shedding rule come into play; synaptogenesis never occurs .', 'The biological meaning of SSN is worth a comment since it might not be obvious .', 'Our interpretation of SSN as modeled here is a very early phase of very fast synaptogenesis ( i . e . , a large γ ) in the absence of any Hebbian or anti-Hebbian modification , i . e . , ε = 0 .', 'Of course , γ has several physical possibilities , including a growth factor-dependent rate of probing axonal growth or a value reflecting the availability of postsynaptic territory on which to form new synapses .', 'With small alteration , the approach here can be melded with BCM theory .', 'For example , one can use the combination of SSN , the BCM synaptic modification equation , and the shedding principle .', 'However , expanding upon adaptive synaptogenesis with the Δw equation used here holds our interest .', 'In terms of the generality of adaptive synaptogenesis , the reader is cautioned that the specific theory presented here will benefit from , or even require , certain modifications .', 'First , not all synapses of forebrain cortical systems are expected to use synapse overproduction as studied here .', 'Second , different classes of synapses will have different sets of modification rules , and in such cases , the ideas here will have to be modified for successful application .', 'For example , inhibitory and monoamine synapses require different considerations .', 'Similarly , it can be questioned whether this theory applies to', '( a ) excitatory feedback synapses ( perhaps half the synapses of cortex ) or to', '( b ) lifelong associatively dynamic encoders ( e . g . , neurons of the hippocampal system [51] or the short-term memory system of dorsal medial prefrontal cortex ) .', 'That is , the neurons and synapses of such regions present a particular challenge; because the neurons of these regions are constantly changing their identity ( i . e . , the stimulus to which they best respond ) , they require a more nuanced definition of quasi-stability compared to cortical region V1 .', 'One particular result , the excessive firing-rate—that is an observed concomitant of overly rapid synaptogenesis—will be easy to control in a more comprehensive model .', 'For example , and still using adaptive processes , mechanisms such as a local , adaptive threshold-adjustment or some forms of inhibition , with their own set of synaptic modification rules , can be used to force firing-rates to some prescribed level .', 'As one form of such inhibition , lateral inhibition might succeed in improving discriminability of categorical representations .', 'Of course , adding lateral inhibition in the form that exists in neocortex only seems sensible when the input has an appropriate topological property ( which can be added to adaptive synaptogenesis , as found in our own preliminary results; see also Ooyen and van Pelt [15] ) .', 'But still topology , and thus , lateral inhibition are vexing issues when considering clusters defined in high dimension as opposed to the rather simple computations involved in early sensory processing where spatial clustering is inherent ( e . g . , retinotopy , tonotopy , etc . ) .', 'Again considering either inhibition or local threshold modification , these enhancements are less than cure-alls: neither of these mechanisms will affect the synapse overgrowth and the associated adult energy costs; secondly , it is not at all obvious that these mechanisms can recover proportional neuronal allocation when there is the overabundance of synapses as occurs along with very fast development; and then there is the problem of piling too many adaptive processes on top of each other while preserving the existence of quasi-stability .', 'Only in regard to this last objection is there a simple solution , a developmental critical period , which times-out synaptogenesis , i . e . , modulates γ to zero .', 'When expanding a model , one must always check to see that the base functions are not lost .', 'In this regard , it seems sensible to alter the present model when incorporating mechanisms that can by themselves control firing-rates .', 'That is , to incorporate inhibition or local threshold modification forces us to backtrack a little and use a synaptogenesis control-mechanism closer to our original approach [16 , 23 , 27] .', 'Specifically , if one uses other mechanisms to control firing rate E[Zj] , then adaptive synaptogenesis should be curtailed based on the moving average of internal excitation Yj , not based on the moving average of firing-rate Zj .', 'Thus , on top of a Yj shutoff-control of synaptogenesis , there can be an additional adaptive process without the stabilization problem mentioned above .', 'Another advantage of controlling synaptogenesis with the average value of net excitation is that this approach optimizes information-rate and the associated synaptic energy-use ( see [10] Theory section ) .', 'Then separately , other adaptive algorithms can control firing rate , which itself figures into the large energy-cost of long-distance communication .', 'Preliminary simulations of Yj–based synaptogenesis indicate that the major relationships shown here still hold; that is , small overproduction of synapses is good for fast development , but even faster development results in energetically costlier neurons in the sense of acquiring additional excitatory synapses without any benefit to function .', 'A final enhancement of the current model , and one that we consider rather urgent , is a control mechanism for the rate constants ε and γ .', 'The success of the current model and many other computational models of neural network development depend on the ergodic mixing characteristics of the input .', 'This assumption seems sensible prenatally and early postnatally .', 'However , the stimuli of the postnatal environment , particularly in the context of high-level , neocortical processing , are far from ergodic .', 'Thus , upgrading the model to handle the absence of input ergodicity is , for us , an active area of research .', 'Relative to brain function , the assumptions underlying the goals of any empirically observed developmental process ( e . g . , synaptogenesis ) is bound to be contentious .', 'However , we submit that the final arbiter will include the perspective of evolution through natural selection rather than merely the perspective of neuroscience alone .', 'Such a broadened approach can benefit the priorities of empirical neuroscience .', 'For example , there may be a relatively small number of genes that control SSN or γ , making these variables a tempting target for empirical research .'], ['This study uses adaptively constructed , feedforward networks of McCulloch-Pitts neurons [52] .', 'For the synapses that exist , an input vector x ( t ) at time-step t produces internal excitation yj ( t ) of neuron j with weights wij ( t ) .', 'yj ( t ) =∑ixi ( t ) ⋅wij ( t )', 'where xi ( k ) ∈ {0 , 1} and wij ( k ) > 0 . If the internal excitation is greater than a predetermined threshold θ , neuron j produces an output firing .', 'That is ,', 'zj ( t ) ={1ifyj ( t ) >θ , and0otherwise}', 'No interaction exists between the outputs of these neurons ( i . e . there is no feedback or lateral inhibition ) , and each neuron develops its connections independently of all other neurons .', 'There are three parts to adaptive synaptogenesis: synaptogenesis , associative synaptic modification , and synaptic shedding .', 'Synaptogenesis , when allowed to occur , is a random Bernoulli process with parameter γ and is governed by rj ( k ) , a neuron j’s receptivity for synaptogenesis at the conclusion of block k .', 'A block consists of a complete set of randomized inputs ( see Timescales ) .', 'This receptivity depends on a constant γ and the moving average of neuron j’s firing-rate .', 'The moving average zj¯ ( k ) is updated after each block k of input presentations .', 'zj¯ ( k ) =zj¯ ( k−1 ) ⋅α+ ( 1/nt ) ∑t=1ntzj ( k , t ) ⋅ ( 1−α )', 'where α = 0 . 25 , nt is the total number of exemplars in a block .', 'Neurons that fire above rate ρ will not add new synapses .', 'When a new synapse is formed , its initial weight is set to 0 . 1 , an arbitrarily chosen value greater than the shed weight .', 'Synaptogenesis is allowed after each presentation of a block of input patterns .', 'rj ( k ) ={γifzj¯ ( k ) <ρ , and0otherwise}', 'where ρ = 0 . 1 Associative synaptic modification [10 , 17] alters weights and captures correlations in the inputs .', 'Given a binary input vector x ( t ) such that xi ( t ) ∈ {0 , 1} and rate constant ε , where t advances one with each time-step .', 'Synaptic shedding [25] occurs when the weight of a synapse is below 0 . 01 .', 'Because a de novo synapse is given a weight of 0 . 1; it is not immediately subject to shedding .', 'No negative weights exist .', 'In the adaptive synaptogenesis model , there are three timescales ( from shorter to longer duration ) :', '1 ) the presentation of an input pattern ( i . e . one time-step t ) , to which a neuron fires or not;', '2 ) the synaptic modification of existing synapses as determined by the rate constant ε; and', '3 ) synaptogenesis and shedding .', 'Synaptic modification occurs after each presentation of an input pattern , and synaptogenesis and shedding are allowed to occur after presentation of a block k of input patterns .', 'The rate constants , ε and γ , determine the exact timescales of synaptic modification , shedding , and synaptogenesis .', 'In one block , all input vectors are presented to the network in a randomized order .', 'An input block consists of a sequence of randomly chosen prototypes that are each randomly perturbed .', 'The random prototype selection is well-mixed .', 'For each postsynaptic neuron , input blocks are presented until no synapses are gained or lost for 200 blocks , at which point a neuron’s synapses are assumed stable .', 'At this time , the weights have achieved their values predicted by the adaptive synaptogenesis theory [10] .', 'See the section Input datasets for the block sizes for each input dataset .', 'For the purpose of the study , values for parameters ε and γ are manipulated ( see Adaptive Synaptogenesis in Methods ) .', 'The parameter values are listed in Table 1 . Table 2 shows assumptions and simplifications of the model used here .', 'At each value of γ , there is a best value of ε that produces fast convergence to stability .', 'For Dataset A , such ε settings were found by sweeping over a range of values and creating the empirical convex functions as seen in Fig 8A .', 'See S1 Appendix for further remarks concerning this convexity .', 'For γ values below 0 . 001 and the input sets used here , the best ε value is approximately predicted by a fixed ratio ( ca . 5 ) between ε and γ .', 'For γ values above 0 . 001 , the relationship breaks down; at such γ values , an ε setting greater than 0 . 004 produces neurons that develop slower ( Fig 8B ) .', 'At even higher γ values of 0 . 01 , neuron allocation is not proportional to category frequency ( See Neuron Allocation in Results ) .', 'For Dataset C , the values of ε and γ are both 0 . 001 .', 'There are two classes of input environments studied here: ( A and C; Table 3 , S3 Code ) .', 'Datasets A1 and A2 ( Fig 9A and 9B ) have five orthogonal prototypes , each defined by 200 adjacent input lines valued one ( i . e . , “on” ) and the remaining are valued zero ( i . e . , “off” ) .', 'In Dataset A1 , the prototypes have relative frequencies of 0 . 1 , 0 . 15 , 0 . 2 , 0 . 25 , and 0 . 3 for a total of 100 input patterns per block .', 'Dataset A2 has equally probable categories .', 'Random noise is added to our input environments .', 'In Dataset A1 , 100 input lines of the prototype are turned off ( 100 off-noise ) and 100 input lines not of the prototype are turned on ( 100 on-noise ) .', 'For each input pattern , the total number of active input lines remains constant .', 'For Dataset A2 , we manipulated the level of on- and off-noise to investigate the effects of noise in the input environment on neuronal development ( See Noise-to-signal ratio constrains best ε ) .', 'The C datasets crudely mimic the input environment of orient
['Introduction', 'Results', 'Discussion', 'Methods']
['Neural circuit development requires that synapses be formed between appropriate neurons .', 'In addition , for a hierarchical network , successful development involves a sequencing of developmental events .', 'It has been suggested that one mechanism that helps speed up development of proper connections is an early overproduction of synapses .', 'Using a computational model of synapse development , such as adaptive synaptogenesis , it is possible to study such overproduction and its role in speeding up development; it is also possible to study other outcomes of synapse overproduction that are seemingly new to the literature .', 'With a fixed number of neurons , adaptive synaptogenesis can control the speed of synaptic development in two ways: by altering the rate constants of the adaptive processes or by altering the initial number of rapidly but non-selectively accrued synapses .', 'Using either mechanism , the simulations reveal that synapse overproduction appears as an unavoidable concomitant of rapid adaptive synaptogenesis .', 'However , the shortest development times , which always produces the greatest amount of synapse overproduction , reduce adult performance by three measures: energy use , discrimination error rates , and proportional neuron allocation .', 'Thus , the results here lead to the hypothesis that the observed speed of neural network development represents a particular inter-generational compromise: quick development benefits parental fecundity while slow development benefits offspring fecundity .']
['The brain processes information by building and adjusting neural connections , and as an organism learns about the environment , it incorporates associations between features of the environment into these adjustable connections .', 'Organisms , however , are not born with these connections; synapses must develop from a connectionless set of neurons .', 'An important aspect of normal brain development is the timely development of the brain and of hierarchically arranged brain regions .', 'In this paper , we show that an unsupervised algorithm , called adaptive synaptogenesis , builds neurons with stable connections and allows these neurons to appropriately discriminate patterns .', 'Here , we show that larger rates of synaptogenesis and synaptic modification speed up development of stable connections .', 'However , such a speed-up produces larger , steady-state energy-costs .', 'We conjecture that these synaptic modification rates have evolved as a compromise via natural selection .', 'In this regard , the evolution-theoretic idea of parent-offspring conflict arises as a developmental consideration; that is , time to stable connectivity must balance with subsequent neural performance .', 'Importantly , we illustrate that the widely-observed phenomenon of synapse overproduction during development can be understood as a compromise between speedy and efficient development .']
['medicine and health sciences', 'action potentials', 'engineering and technology', 'nervous system', 'membrane potential', 'electrophysiology', 'neuroscience', 'physiological processes', 'systems science', 'mathematics', 'prototypes', 'computer and information sciences', 'ergodicity', 'developmental neuroscience', 'molting', 'animal cells', 'technology development', 'cellular neuroscience', 'cell biology', 'anatomy', 'synapses', 'data visualization', 'physiology', 'neurons', 'biology and life sciences', 'cellular types', 'physical sciences', 'neurophysiology']