key: cord-0485134-zl2ltoqj
authors: Tao, Jun-Bo; Sun, Bai-Qing; Zhu, Wei-Dong; Qu, Shi-You; Chen, Ling-Kun; Li, Jia-Qiang; Wu, Chong; Xiong, Yu; Zhou, Jiaxuan
title: Plasticity Neural Network Based on Astrocytic Influence at Critical Period, Synaptic Competition and Compensation by Current and Mnemonic Brain Plasticity and Synapse Formation
date: 2022-03-19
journal: nan
DOI: nan
sha: da09d7bcddcb7927873ce74e72c072b995cf0d7f
doc_id: 485134
cord_uid: zl2ltoqj

The mechanism of our NN is in line with the results of the latest MIT brain plasticity study, in which researchers found that as a synapse strengthens, neighboring synapses automatically weaken themselves to compensate. Regarding this mechanism, Dr. Luo's team at Stanford University has put forward that competition regarding synapse formation for dendritic morphogenesis is crucial. The astrocyte impacts on Brain Plasticity and Synapse Formation is an important mechanism of our NN at critical period and closure of critical period. We try to conduct research of failure in brain plasticity by model at the closure of critical period by contrasting with studies before. Cutting edge imaging and genetic tools are combined in their studies, whereas our research lays more emphasis on a new NN. In tests, possible explanations of dendrite morphogenesis are derived, which demonstrate that dendrite generation, to a certain extent, is curbed by synapse formation. Current and mnemonic brain plasticity as well as synaptic action range are also taken into account in the study. Furthermore, the frame of NN is based on current and mnemonic dynamic gradient informational synapse formation. The mnemonic gradient information needs to take into account the forgotten memory-astrocytic synapse formation memory factor. Mnemonic brain plasticity involves the plus or minus disturbance-astrocytic brain plasticity phagocytose factor. The influence of astrocyte made local synaptic action range remain in an appropriate length at critical period. Through the tabular data of the PNN test, we found that the memory factor of astrocytes, like the phagocytose factor, produces the effect of reducing the local accumulation of synapses. Therefore, is it possible to reduce the number of animal experiments and their suffering by simulating and planning the factors of biological experiments through Deep Learning models?

Plasticity Neural Network Based on Astrocytic Influence at Critical Period, Synaptic C Heilongjiang 150001, China. Email: 1191000315@stu.hit.edu.cn * Correspondence and requests for materials should be addressed to Chong Wu (email: nnem2022@outlook.com) and Yu Xiong(email: y.xiong@surrey.ac.uk) * * These authors contributed to the work equally and should be regarded as co-first authors.

Beyond the consideration of the basic connection, this research focus on a new Neutral Network (NN) model, referred as PNN (Plasticity Neural Network) in which the current and mnemonic range of action between synapses and neurons plastic changes with iteration are both taken into account at the same time, specifically, the current and mnemonic synaptic range of action weights. In addition, the coding result on PNN mode is given that as one synapse strengthens, neighboring synapses weaken on their own to compensate it; this mechanism is consistent with the findings in the recent research on brain plasticity at the MIT Picower Institute. Regarding the importance of this mechanism, Dr. Luo's team at Stanford University has mentioned that the competition regarding synapse formation for dendritic morphogenesis is important. The influence of astrocyte impacts on brain plasticity and synapse formation is an important mechanism of our Neural Network at critical periods and the end of critical periods. We try to examine the mechanism of failure in brain plasticity by model at the end of critical periods in details by contrasting study before [17] . The new PNN model is not just modified on the frame of NN based on current gradient informational synapse formation and brain plasticity, but also the mnemonic gradient informational synapse formation and brain plasticity at critical periods. The mnemonic gradient information needs to consider forgotten memory-astrocytic synapse formation mnemonic factor. And mnemonic brain plasticity involves plus or minus disturbance-astrocytic brain plasticity disturbance factor. The influence of astrocyte made local synaptic range of action remains an appropriate length at critical periods. When one synapse transfers a signal from one neuron to other with a better stimulation signal, PNN will change the current synaptic range of action based on Mean Squared Error (MSE) of loss function, and vice versa for the same reason. For a given neuron, the synaptic plasticity of the connecting neurons from input to output units is enhanced and diminished with iteration. Considering the Recurrent Neural Network (RNN), each input variable corresponds to neurons sharing connection weights over a period of time interval; the weights are updated by the MSE loss of the activation function of the amount of output change within this synaptic range of action. The synaptic range of action is reflected in the time series, for which the real value of the simulation would lead to gradient-based change in the synaptic range of action by Back Propagation. The Back Propagation of PNN includes connection weights and synaptic range of action weights. In the simulation, the synaptic position is fine-tuned in every iteration. The range of synaptic changes cannot be too large, which must retain within a minimum value for each synaptic formation, thus making the synaptic range of action more sensitive to the plasticity than the elasticity. The cosine filter is used in the tests for comparing the result of the three cases, namely gradient optimized synaptic range (ORPNN), random synaptic range (RRPNN), and constant synaptic range (CRPNN). The result indicates that the OPRNN runs the best. The iteration of CRPNN method indicates difficult to converge. RRPNN method is slower convergent than ORPNN method on finding the optimal solution. Considering the insights from the practical pandemic, we attempted to explain the PNN model. A simulation of PNN was conducted on the cognitive processes in the brain from infancy to senile phase, which is interpreted by the observation of decreasing the synapse population and increasing the minimum of synaptic range of action in PNN evolution of synapse formation, hence the loss of diversity and plasticity of cognition. This explanation is similar to the Dr. Luo and his colleagues' research, and on PNN which shows that the synapse formation in a certain extent may inhibit dendrite. And we also analyzed the different brain plasticity by tables at critical and the end of critical periods. Furthermore, the RNN-based PNN formula is simple and easy to implement; and the contribution to NN is as seminal as BP. Most of our simulations did not refer the biological tests and papers besides hypothesis of Dr. Luo's team in synapse formation, and our question is if model construction, formula derivation and algorithm testing to promote the development of neuroscience and brain cognition? And it has no ethics of animal testing.

Keywords: Deep learning; Synaptic Plasticity; Synaptic compensation; Synaptic competition; Constant synaptic action range; Random synaptic action range; Optimization synaptic action range; PNN; RNN; BP; Computational neuroscience; current and mnemonic brain plasticity; current and mnemonic gradient informational synapse formation; astrocytic synapse formation memory factor; astrocytic brain plasticity phagocytose factor; critical period

In brief: The mechanism of our proposed Neural Network (NN) is very well in line with the results of the latest MIT brain plasticity study, in which researchers found that as a synapse strengthens, neighboring synapses automatically weaken themselves to compensate [13, 14] . Regarding the importance of this mechanism, Dr. Luo's team at Stanford University has put forward that competition regarding synapse formation for dendritic morphogenesis is crucial [15] [16] [17] . The influence of astrocyte impacts on Brain Plasticity and Synapse Formation is an important mechanism of our Neural Network at critical period and closure of critical period. We try to conduct research on the mechanism of failure in brain plasticity by model at the closure of critical period in details by contrasting with studies before [18] . Cutting edge imaging and genetic tools are combined in their studies, whereas our research lays more emphasis on the model, derivation and simulation of a new NN. In tests, possible explanations of dendrite morphogenesis are derived, which demonstrate that dendrite generation, to a certain extent, is curbed by synapse formation [15] [16] [17] . Current and mnemonic brain plasticity as well as synaptic action range are also taken into account in the study. Furthermore, the frame of the new NN is based on current and mnemonic dynamic gradient informational synapse formation. The mnemonic gradient information needs to take into account the forgotten memory-astrocytic synapse formation memory persistence factor [19] . Mnemonic brain plasticity involves the plus or minus disturbance-astrocytic brain plasticity phagocytose factor [20] . The influence of astrocyte made local synaptic action range remain in an appropriate length at critical period [20] . Through the tabular data of the PNN test, we found that the memory persistence factor of astrocytes, like the phagocytose factor, produces the effect of reducing the local accumulation of synapses. Therefore, is it possible to reduce the number of animal experiments and their suffering by simulating and planning the factors of biological experiments through deep learning models?

Summary: Beyond the consideration of the basic connection, this research focuses on a new Neutral Network (NN) model, referred as PNN (Plasticity Neural Network) in which the current and mnemonic range of action between synapses and neurons plastic changes with iteration are both taken into account at the same time, specifically, the current and mnemonic synaptic action range weights. In addition, the mechanism of synaptic compensation; this mechanism is consistent with the findings in the recent research on brain plasticity at the MIT Picower Institute. Regarding the importance of this mechanism [13, 14] . Dr. Luo's team at Stanford University has mentioned that the competition regarding synapse formation for dendritic morphogenesis is important. The influence of astrocyte impacts on brain plasticity and synapse formation is an important mechanism of our Neural Network at critical period and the closure of critical period . We try to examine the mechanism of failure in brain plasticity by model at the closure of critical period in details by contrasting with studies before [18] . The PNN model is not just modified on the frame of NN based on current gradient informational synapse formation and brain plasticity, but also the mnemonic gradient informational synapse formation and brain plasticity at critical period. The mnemonic gradient information needs to consider the forgotten memory-astrocytic synapse formation memory persistence factor [19] . In addition, mnemonic brain plasticity involves the plus or minus disturbance-astrocytic brain plasticity phagocytose factor [20] . The influence of astrocyte made local synaptic action range remain in an appropriate length at critical period [20] . When one synapse transfers a signal from one neuron to other with a better stimulation signal, PNN will change the current synaptic action range based on Mean Squared Error (MSE) of loss function, and vice versa for the same reason. For a given neuron, the synaptic plasticity of the connecting neurons from input to output units is enhanced and diminished with iteration. Considering the Recurrent Neural Network (RNN), each input variable corresponds to neurons sharing connection weights over a period of time interval, and each time interval corresponds to a connection weight; the weights are updated by the MSE loss of the activation function of output within this synaptic action range. The synaptic action range is reflected in the time interval, for which the real value of the simulation would lead to gradient-based change in the synaptic action range by Back Propagation. The Back Propagation of PNN includes connection weights and synaptic action range weights. In the simulation, the synaptic position is fine-tuned in every iteration. The range of synaptic changes cannot be too large for each synaptic formation. The cosine filter is used in the tests for comparing the result of the three cases, namely gradient optimized synaptic action range (ORPNN), random synaptic action range (RRPNN), and constant synaptic action range (CRPNN). The result indicates that the OPRNN runs the best. The iteration of CRPNN method indicates difficult to converge. RRPNN method is slower convergent than ORPNN method on finding the optimal solution. Considering the insights from the practical pandemic, we attempted to explain the PNN model. A simulation of PNN was conducted on the cognitive processes in the brain from infancy to senile phase, which is interpreted by the observation of decreasing the synapse population and increasing the minimum of synaptic action range in PNN evolution of synapse formation, hence the loss of diversity and plasticity. This explanation is similar to the Dr. Luo and his colleagues' research, and on PNN which shows that the synapse formation in a certain extent may inhibit dendrite. And we also analyzed the different brain plasticity by Tables at critical and the closure of critical period. When ORPNN contains both astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor will have better results in correlation coefficients of Tables respectively at critical period. Through the tabular data of the PNN test, we found that the memory persistence factor of astrocytes, like the phagocytose factor, produces the effect of reducing the local accumulation of synapses. Therefore, is it possible to reduce the number of animal experiments and their suffering by simulating and planning the factors of biological experiments through deep learning models? Furthermore, the RNN-based PNN formula is simple and easy to implement; and the contribution to NN is as seminal as BP. Most of our simulations did not refer to the biological tests and papers besides the hypothesis of Dr. Luo's team in synapse formation, and our question is can model construction, formula derivation and algorithm testing serve to promote the development of neuroscience and brain cognition?

Researchers in the Deep Learning community have long regarded simulation of the human brain as an important means of advancing Neural Networks (NN). In machine learning and computational neuroscience, Artificial Neural Networks (ANN) are used as mathematical and computational models that mimic the structure and function of biological Neural Networks. Estimations and approximations conducted by the functions have witnessed great success over the recent decades. The more commonly used Neural Networks models today include Perceptron in 1957 [1] , Hopfield network in 1982 [2] , Boltzmann in 1983 [3] , Back Propagation (BP) in 1986 [4] , Convolutional Neural Networks (CNN) in 1989 (from Neocognitron [5] to CNN [6] ), Spiking Neural Networks (SNN) in 1997 [7] , Long Short-Term Memory (LSTM) in 1997 [8] , Deep Belief Network (DBN) in 2006 [9] , Deep Neural Networks (DNN) in 2012 [10] and Deep Forest (DF) in 2019 [11] . Human brains can be very flexible due to their plasticity. Neural activity can induce the strengthening or weakening (potentiation or depression) of synapses to produce plasticity. The brain plasticity therefore refers to the connections of synapses and neurons, and to establish new connections as a result of learning and experiencing, thus exerting impacts on the behaviors of individuals. The synapses are the roles between neurons permitting the transmission of signals or stimulus. Therefore, our new NN model takes into account not only the synaptic connections, but also how the synaptic action range at current and historical moments changes plastically with the iteration, particularly at critical period, namely, in addition to the weights of the current and mnemonic synaptic connection, it also considers the weights of the current and mnemonic synaptic action range (brain plasticity) and gradient information (synapse formation) at critical period.

Last year, Dr. Fei-Fei Li presented the One Hundred Year Study on Artificial Intelligence (AI100) 2021 Study Panel Report [12] , in which one question concerns how much progress we have made in UNDERSTANDING the PRINCIPLES of HUMAN INTELLIGENCE. First, a basic tenet of cognitive neuroscience is that individual characteristics such as working memory and executive control are critical for domain-independent intelligence, which determines an individual's performance on all cognitive tasks regardless of mode or subject. A second hypothesis gaining traction is that higher-ability persons enjoy more efficient patterns of brain connectivity-higher-level brain regions in the parietal-frontal cortex. The third concept is more revolutionary, which suggests that the neural correlates of intelligence are dispersed throughout the brain. According to this theory, human intelligence is fundamentally flexible, constantly updating past information and making new predictions [12] . The new NN proposed in this thesis is grounded on the third notion, taking into consideration the current and past brain plasticity and gradient informational synapse formation at critical period.

Scientists of MIT's Picower Institute have unveiled for the first time how this balance is achieved in synapses: Professor Mriganka Sur likens this behavior to a massive school of fish in the sea. Immediately when the lead fish changes direction, other fish will follow suit, presenting a delicate marine "dance". "Collective behaviors of complex systems always have simple rules. When one synapse goes up, within 50 micrometers there is a decrease in the strength of other synapses using a well-defined molecular mechanism," the scientists stated [13, 14] . The lead behavior of the school of fish is well embodied in our PNN.

Findings largely consistent with that of Dr. Luo's team at Stanford University in PNN tests are obtained in our study, that is, the synapse formation will inhibit dendrites generation to a certain extent [15] [16] [17] , with the reason shown in our simulation results being that synaptic growth leads to a reduction in changes in synaptic action range, which disrupts diversity and plasticity of brain.

The finding has significant implications for the learning and further development of NN. Moreover, the mechanism of our NN is basically consistent with that of the latest brain plasticity study conducted by MIT, in which researchers found that as a synapse strengthens, neighboring synapses automatically weaken themselves to compensate [13, 14] . In tests, possible explanations of dendrite morphogenesis are derived, which demonstrate that dendrite generation, to a certain extent, is curbed by synapse formation [15] . Unconventional astrocyte connexin signaling hinders expression of extracellular matrix-degrading enzyme matrix metalloproteinase 9 (MMP9) through RhoA-guanosine triphosphatase activation for controlling critical period closure. The influence of astrocyte impacts on brain plasticity and synapse formation is an important mechanism of our Neural Network at critical period, and failures by the closure of critical period result in neurodevelopmental disorders [18] . In the model at critical period, the hypothesis is the optimal brain plasticity so far affects current brain plasticity and the optimal synapse formation so far affects current synapse formation. While their research features a combination of cutting-edge imaging and genetic tools, our study lays more emphasis on the model, derivation and simulation of a new NN. At the same time, the current and mnemonic brain plasticity-the synaptic action range are taken into consideration. Furthermore, the frame of new NN is based on current and mnemonic gradient informational synapse formation.

Our newly proposed NN is named the Plasticity Neural Network (PNN). For a given neuron, the synaptic plasticity of the connecting neurons from input to output units is first enhanced and then diminished with iteration. Synaptic competition causes the enhancement of signal-stimulated synapses to be reflected in an increase in synaptic action range, while the weakening of peripherally stimulated synapses is reflected in a shortening of synaptic action range.

The following papers are academic frontiers in neuroscience which we endeavor to make connections with our PNN.

Their research shows that SynCAM 1 actively limits cortical plasticity in the mature brain. Plasticity tapers off when the brain matures and the conclusion is sufficiently substantiated by visual input in adult animal models [20] .

Their research shows that SynCAM 1 actively limits cortical plasticity in the mature brain. Plasticity tapers off when the brain matures and the conclusion is sufficiently substantiated by visual input in adult animal models [19] .

The Self-Back Propagation (SBP) phenomenon, first discovered in hippocampal neurons [23, 27] , involves cross-layer Back Propagation (BP) of Long-Term Potentiation (LTP)and Long-Term Depression (LTD) from output to input synapses of a neuron to enable the strengthening or weakening of synaptic connections. Other forms of nonlocal spreads of LTP and LTD in the pre-and post-synaptic neurons have already been researched extensively [23, 24, 27] . The SBP phenomenon induces a form of nonlocal activity-dependent synaptic plasticity that may endow developing neural circuits with the capacity to modify the weights of input synapses on a neuron in accordance with the status of its output synapses [24] . The existence of SBP was demonstrated in developing retinotectal circuits in vivo [25, 26] . These papers endeavor to shed light on local plasticity rules involving activity-dependent modification of synapses to obtain synaptic activity [23] [24] [25] [26] [27] .

Bertens and Lee proposed an evolvable neural unit (ENU) that can evolve individual somatic and synaptic compartment models of neurons in a scalable manner and try to solve a T-maze environment task [28] .

Wang and Sun stated that in a unidirectional RNN, the linking with the emotion regions and the somatic motor cortex includes three basic units: input units arriving from the emotion regions, only one hidden unit composed of selffeedback connected medial prefrontal cortex neurons, and output units located at the somatic motor cortex [29] .

In the simulations of PNN in brain plasticity at critical period, we had an idea to modify the synapses for repairing brain plasticity, and later we found a paper to build and remodel of synapses [30] .

Instead of conducting research simply on the basis of Self-Back Propagation phenomenon, the PNN we propose in this thesis focuses more on plasticity and compensation as a result of its tapering off -the brain from infancy to senile phase during critical period and by the closure of critical period . The model also delves into the competition of brain plasticity which introduces a novel form of current and mnemonic gradient informational synapse formation, as well as current and mnemonic brain plasticity at critical period. The strengthening and weakening of synaptic connection help to curb and boost current and historical synaptic action range, resulting in an increased accuracy of PNN.

The research, based on Genetic Algorithm or Particle Swarm Optimization in search of the connection weights for each time interval of the dynamic problems, and on PNN to find the weights of the connection and the range of action, takes into account the specificity of the new coronavirus variant, whose viral strain is highly variable. the viral load of the Delta (or Omicron) strain is 1,260 times higher than the prevalent strain of last year, thus doubling the infectious rate of last year's original strain. The major difference between PNN and RNN is that the former has a formula to estimate the parameters (connection weights) for their respective parameters' ranges (synaptic action range weights) at critical period. So PNN is based on the practical dynamic problem in an effort to obtain the new NN. As for the selection of the appropriate mechanism, we chose the hypotheses and possible explanations of synaptic compensation, competition and influence of astrocyte [13] [14] [15] [16] [17] [18] . In this way, not only the frame of NN is transformed by current gradient informational synapse formation and brain plasticity, but also mnemonic gradient informational synapse formation and brain plasticity is taken into account. The mnemonic gradient information needs to consider astrocytic synapse formation memory persistence factor [19] .In addition, mnemonic brain plasticity involves the astrocytic brain plasticity phagocytose factor [20] .

After representing the synaptic plasticity competition in Formula (2), compensation in Formula (3), current and mnemonic gradient informational synapse formation at critical periods in Formula (2), current and mnemonic brain plasticity at critical periods in Formula (4), cosine filter is used in tests to verify the PNN and compare the results of the following three situations: the case dubbed as CRPNN, in which the synaptic action range for various neurons remains unchanged; the RRPNN case, in which synaptic action range for each neuron generates in a random manner; the last one is the ORPNN case, in which iterative optimization of synaptic action range connects neurons over time. The findings also show that synapse formation will, to some extent, inhabit the tests of PNN. And we also analyzed the different brain plasticity by Tables at critical period and the closure of critical period. The influence of astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor in Tables.

We developed PNN under the premise that the total synaptic action range of these neurons' connections remain a constant value for the synaptic compensation. For RNN, the size of training sets has been determined by total synaptic action range l max . However, synapse formation will lead to convergence and the loss of plasticity, therefore diminishing the cognitive abilities of PNN. This hypothesis works for RNN as well as BP. For RNN, each input variable corresponds to the weight of the neurons' shared connection within a time interval, and each time interval corresponds to a connection weight. and the weight is updated by the MSE loss of the activation function of output within this synaptic action range. The synaptic action range is reflected in the time interval, and alters with the gradient by simulating real value-training sets in iteration.

Formula (1) and (2) are both obtained through the reasoning of MSE loss function gradient. The activation function for the amount of change in the output result is processed through the tanh function.

Formula (1) updates the connection weight of the kth synapse of the mth input neuron, f () represents the output of different variables and synapses, m represents the label of the input variable, k represents label of synapses connecting two neurons, and the range of values of time series t is [n3(m, k) n3(m, k+1)]. h(t) indicates actual output, and f (t) represents desired output. n3(m, k) and n3(m, k+1) depict the location of the mth input variable and kth synaptic initiation and termination points. η = 1/1E3 × (j max ) − j, namely, the learning rate is reflected in formulas (1) and (2).

The synaptic range of action weights of the mth input variable and the kth synapse is updated by Formula (2) . The relationship between the synaptic location and the range of action satisfies the equation n 3 (m, k+1)−n 3 (m, k) = n 1 (m, k), in which n 1 (m, k) represents the synaptic action range. When the synaptic position of the lead neurons is changed, these neurons, much like the heads in a massive school of fish, also exerts an impact on the synaptic position of the posterior neurons. Formula (2) considers current and mnemonic gradient informational synapse formation updated by the optimal gradient of synaptic action range weight so far [g r (m, k)] best , so that frame of PNN bears current and mnemonic gradient information at critical period. With reference to Formula (3), Formula (2) (n(m, k) 1 ) r(m,k) = (r(m, k) × (−1)/sum(r(m, : )) 2 ) + 1/sum(r(m, :)) 2 × (l max − k max × l min ).

r(m, k) = r(m, k) − g r (m, k) or r(m, k) = r (m,k) − g r (m, k) + (g r (m, k) best − g r (m, k)) × M Formula (3) serves to update the range of connecting neurons at the kth synapse of the mth input variable, and normalization of result is achieved through r(m, k)/sum(r(m, :)) . The result of r (m,k) = r (m,k) − g r (m, k) + (g r (m, k) best − g r (m, k)) × M may be better than r(m, k) = r(m, k) − g r (m, k) in simulation. If Formula (3) just has current gradient information, r(m, k) = r(m, k) − g r (m, k), else if Formula (3) includes current and mnemonic gradient information, r(m, k) = r (m,k) − g r (m, k) + (g r (m, k) best − g r (m, k)) × M needs to consider forgotten memory in synapse formation.

The derivation of memory persistence factor is as follows: The update of r(m, k) = r(m, k) − g r (m, k) is based on the gradient descent method. By r(m, k) = r(m, k) − g r (m, k) best , we get r(m, k) (t+1) = r(m, k) t − g r (m, k) best , r(m, k) (t+1) is the optimal synaptic action range weight so far, and g r (m, k) best is the optimal gradient of synaptic action range weight so far. Time reversal from t + 1 to t to return to in situ to maintain memory can also be written as r(m, k) t = r(m, k) (t+1) + g r (m, k) best , after which, by adding r(m, k) (t+1) = r(m, k) t −g r (m, k) and r(m, k) t = r(m, k) (t+1) +g r (m, k) best , we get r(m, k) = r(m, k)−0.5×g r (m, k)+0.5×g r (m, k) best .Considering the flexibility of learning rate, it can also be written as r(m, k) = r(m, k) − g r (m, k) + (g r (m, k) best − g r (m, k)) × M .

In the model at critical period, the hypothesis is the optimal synapse formation so far affects current synapse formation. Bad memory of synapse formation F = exp(−6×j/j max )×rand or good memory M = exp(−4.5×j/j max )×rand can be called astrocytic synapse formation memory persistence factor. j/j max illustrates the decreasing weights of factor we present at critical period in M .

For a certain input variable, the neuron has a total of k max synapses,l min shows that the range of synaptic change enjoys a certain degree of plasticity, and the minimum synaptic action range is l min . g r (m, k) controls l min with iterations and represents the elasticity or plasticity of synapses, and if the value of l min is overly small, the change of n 1 (m, k) will be too elastic and may make the test results deteriorate. k max indicates the population of synapses involved in the neurons from input to output units, and l max represents the total range of time series and is a constant. For the signal of a neuron (i.e., an input variable), when one synapse strengthens, there must be surrounding synapses that are weakened. In the case when l max = k max × l min , i.e., the case in which the synaptic action range is constant, the n 1 (m, k) is also a constant value. That is, for l min = l max /k max , constant-range synapses also satisfy Formula (2) and Formula (3). Formula (2) can be named the synaptic competitive formula, and Formula (3) can be dubbed as the synaptic compensational formula.

Formula (4) updates r(m, k) by current and mnemonic synaptic action range weight, the optimal r(m, k) best so far. Formula (4) is named the brain plasticity formula. In the model featuring critical period, the hypothesis is the optimal brain plasticity so far affects current brain plasticity. Fig. 1 astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor P = sign(rand − 0.5) × rand × (j max − j)/j max implements plus or minus disturbance of brain plasticity-astrocytic brain plasticity phagocytose factor. The dynamic forgotten memory astrocytic synapse formation memory persistence factor and disturbance astrocytic brain plasticity phagocytose factor shown in Fig. 1 . (j max − j)/j max illustrates we give decreasing weights of factor at critical period in P . sign(rand − 0.5) gives Formula P both positive and negative results. P can be negative because of astrocytes phagocytose synapses, but considering the compensation of synaptic effect, P should be positive and negative. The result decreases with the maturation response of astrocytes.

The pseudocode of PNN is as follows, which encompasses Forward and Back Propagation. ) and r(m, k) * , the global synaptic action range weights r(m, k) need to be changed by overall optimization, while the same type of connection weight w(g, k) just needs to be changed by local optimization. i.e., research on disease dynamics, when infection rates (or cure rates) change, the time interval of the infection rate and cure rates both need to be updated at the same time.

Parameters regarding the abovementioned test is as follows: testing environment is WIN11 and MATLAB2020a, the parameters for updating the weights remain the same for the three tests, and the number of iterations is j max = 40000. The tests employ cosine filter, in which input are filter from point t + 1 to t + 4, and prediction of output is filter at t + 5 point. l min = 2, k max = 9, l max = 44. Three methods were all considered mnemonic optimum solution, so RRPNN and ORPNN both take into consideration the current and mnemonic weight of connection and synaptic action range-brain plasticity. But CRPNN only pays heed to the current and mnemonic weight of connection, as the synaptic action range remains constant in synapse formation. Only ORPNN takes into account the current and mnemonic gradient informational synapse formation, and it also considers current and mnemonic brain plasticity. Fig.2, Fig.3 , Fig.4 , in which the results of the variation of the logarithm of the loss of PNN with iterations are also provided. Result of the first test: the correlation coefficient between CRPNN simulation data and actual data is 0.8057, the correlation coefficient between RRPNN simulation data and real data is 0.8706, and the correlation coefficient between ORPNN simulation data and real data is 0.9617. Result of the second test: the correlation coefficients are 0.8292, 0.8921 and 0.9626 respectively. Result of the third test: the correlation coefficients are 0.8031, 0.8728 and 0.9606 respectively.

The results of the RRPNN test turned out to be satisfactory, and though the connection weights are updated by equation (1) and mnemonic optimal solution, more satisfactory optimization results could be found for random synaptic action range locations.

The ORPNN showed strong fluctuation in the iterations, whereas the final convergence turned out better and faster than the RRPNN and the CRPNN.

Iteration of CRPNN is not convergent. RRPNN has slow convergence. The number of iterations is j max = 40000. In order to study the impact of the astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor, Fig.5 is obtained. In the first scenario, only the astrocytic brain plasticity phagocytose factor is taken into account with the optimal r(m, k) best so far and a correlation coefficient of 0.9499, ORPNN-PF. The second scenario takes into account both the astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor in formula (2) . It may have better results and a correlation coefficient of 0.9596 at critical period, ORPNN-MF-PF. The third scenario excludes the astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor at the closure of critical period then astrocytes become mature, with a correlation coefficient of 0.9257, ORPNN. The fourth scenario only the astrocytic synapse formation memory persistence factor is taken into account with the optimal gradient g(m, k) best so far and a correlation coefficient of 0.9308, ORPNN-MF. In Fig.5 , PF means brain plasticity phagocytose factor, MF means synapse formation memory persistence factor.

We then modified the ORPNN model accordingly for analysis. By modifying l min , we managed to simulate the process of brain cognition from infancy to the senile phase. l min exhibits elasticity at the beginning as a decimal value, then gradually decreases in plasticity. In this way the increase in synaptic action range from l min = 2 to l max /k max ≈ 5 made diversity of synaptic action range progressively smaller. CRPNN corresponds to the senile phase period of brain cognition as it satisfies l min = l max /k max , and the funny thing is that the iteration is not convergent at CRPNN.

We obtained a possible explanation to the research findings of Dr. Luo's team through PNN. We drew the relationship of different l min (minimum synaptic action range), different k max (synapse population) and correlation, (synapse population) and correlation coefficients, and the results are shown in Fig. 6 below. We also conducted another test, in which the cosine filter and cycle gradually diminish over time-variable cycle cosine filter, and the number of iterations is j max = 80000. TThe functions of dendrites are to receive and process signals from other neurons. Our simulation results reveal that synapse formation will inhibit dendrites to a certain extent, furthering the findings of Dr. Luo's team at the Stanford University. An explanation to this is that synaptic growth will lead to a reduction in the changes in synaptic action range, and an overgrowth will inevitably disrupt the diversity and plasticity of human brain. l min = 2 − 5, k max = 7 − 9. Fig.6 concerns the relationship between synapse formation and correlation coefficients in ORPNN. The synapse formation could involve a decrease in k max and an increase in l min . The PNN situation can be thought as the Finite Element Method, in which Fig. 6 Relationship between synapse formation (different l min and kmax)and correlation coefficients:(a) ORPNN (ORPNN contains neither astrocytic synapse formation memory persistence factor nor astrocytic brain plasticity phagocytose factor at the closure of critical period), (b) ORPNN (both containing astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor at critical period) neurons of the brain can be regarded as nodes of FEM. As increased in the number of nodes and finer meshes can lead to more accurate computational results of FEM, more neurons and smaller minimum synaptic action range can also help us obtain more desirable and accurate PNN results. The ORPNN contains both mnemonic brain plasticity and dynamic gradient informational synapse formation. When synaptic population k max decreases and total synaptic action range l max is a constant, the diversity of local synaptic action range may be improved as the synaptic action range enlarges from l min to l max /k max , so k max decreases may enhance correlation coefficients by accident, as you can observe from Fig. 7 (a) l min = 6, k max = 7 − 8 and l min = 2, k max = 7 − 8.

As for the ORPNN situation (both containing mnemonic brain plasticity and mnemonic dynamic gradient informational synapse formation), synaptic action ranges with different l min values and different input variables are provided in Table 1 . When l min = 5, the synapse grows to a certain extent, and synaptic action range remains almost the same, thus leading to a worsened Fig. 7 ORPNN test results are l min = 2−5 and kmax = 9 respectively (a)ORPNN (ORPNN contains neither astrocytic synapse formation memory persistence factor nor astrocytic brain plasticity phagocytose factor at the closure of critical period) simulation result. When the synapse grows, standard deviation of synaptic action range at the end of the iteration diminishes, and the correlation coefficients decreases to a certain extent when l min = 5. From l min = 2 to l min = 5, synaptic action range gradually loses its diversity and plasticity. The number of iterations is j max = 80000. And the results of synaptic action range and correlation coefficient when iteration equals 10,000, 20,000, 40,000 and 80,000 were also observed, as 4444 in Table 1 represents the synaptic action ranges are all 4, and 0.9587, 0.9638, 0.9599 and 0.9670 are the correlation coefficients when iteration equals 10,000, 20,000, 40,000 and 80,000, respectively.

As for the ORPNN situation (ORPNN both not containing s neither astrocytic synapse formation memory persistence factor nor astrocytic brain plasticity phagocytose factor), synaptic action ranges with different l min values and different input variables are provided in Table 2 .

Comparing with the Tables, the influence of astrocyte made local synaptic action range remain in an appropriate length at critical period, as one can observe in Tables 1, 3 and 4, else some synapses have longer length and disorder, resulting in the alignment of other synaptic action ranges, as one can Fig. 8 ORPNN test results are l min = 2 − 5 and kmax = 9 respectively (b) ORPNN (both containing astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor at critical period) observe in Table 2 . The simulations in Tables similar to that failure to the closure of critical period results in neurodevelopmental disorders, and then astrocytes become mature [18] . We can find better results of correlation coefficients in Tables respectively at critical period, as model of Deep Learning contains both astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor. Table 2 shows that the local accumulation of synapse length is accompanied by an iterative gradient explosion by Fig.7 and Tables. It can be seen from Table 2 that with the growth of the l min and the gradual loss of iterative activity, the calculation of the correlation coefficient in Table 2 also turns out  worse than in other Tables.  For Table 3 which considers only phagocytose factor, synapse formation showed good activity with iterative increase and l min growth. The calculation of the correlation coefficient was slightly worse than that of Table 1 and better than that of Tables 2 and 4. As the probability of local gradient explosion is small when considering phagocytose factor by Fig.7 and Tables, the result of Fig. 9 ORPNN (both containing astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor at critical period). (c) ORPNN (only contains astrocytic brain plasticity phagocytose factor) Table 3 converges even at l min = 5, while that of other Tables converge poorly at l min = 5. Table 4 only deals with the memory persistence factor, which maintains the slow change of synapse length, and the synapse formation shows a worse activity with iterative increase and minimum synaptic action range growth than in Tables 1 and 3 . The iterative gradient explosion phenomenon of the memory persistence factor is worse than that of Tables 1 and 3, and better than that of Table 2 by Fig.7 and Tables.  Tables 3 and 4 respectively deals with scenarios in which neither memory persistence factor nor phagocytose factor has long local synapses. When l min ≈ l max /k max = 44/9 ≈ 5, similar to the case of CRPNN, it can be seen from Tables 1, 2, 3, 4 that the correlation coefficient effect is not so good at this time, and even Tables 1, 2 and 4 cannot converge.

ORPNN contains both astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor. We propose the simulation of l min = 2−5 and k max = 9 respectively, and the correlation coefficients are 0.9670, 0.9603,0.9173 and 0.8956 respectively. ORPNN both not contains neither astrocytic synapse formation memory persistence factor nor astrocytic brain plasticity phagocytose factor at the closure of critical period, and the correlation coefficients are 0.9331, 0.9410, 0.9387 and 0.8906 respectively. ORPNN only contains astrocytic brain plasticity phagocytose factor, and the correlation coefficients are 0.9504, 0.9650, 0.9536 and 0.8986 respectively. ORPNN only contains astrocytic synapse formation memory persistence factor, and the correlation coefficients are 0.9251, 0.9608, 0.9283 and 0.8882 respectively. Different l min impact's simulations can be observed in Fig. 7 . The number of iterations is j max = 80000.

The mechanism of PNN enables the appropriate synapse to change the range of action for dendrite morphogenesis at critical period. When one synapse transfers a signal from one neuron to other with a better stimulation signal, PNN will change the current synaptic action range based on Mean Squared Error (MSE) of loss function, with reference to current and mnemonic brain Table 1 Synaptic action range with different l min values and different variables-brain plasticity (ORPNN contains both astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor at critical period) plasticity and gradient informational synapse formation, and vice versa for the same reason.

In terms of the infectious disease dynamics, the infection rates of primitive strains and Delta strains varies in different time ranges, and the time interval corresponding to the cure rate will not be in line with the time range corresponding to the infection rate, nor will the cure rate correspond exactly to the synaptic action range effect of the infection rate, due to its high dependence on the medical condition.

Explaining in terms of synaptic competition, PNN is well-positioned to calculate the time ranges corresponding to the cure and infection rates, i.e., the synaptic action range, through data processing, and then obtain the appropriate cure and infection rates, i.e., the synaptic connection weight. During the emergency and non-emergency period, astrocytic influence is similar to government policies such as lockdown and mask rules, thereby affecting infection rate, cure rate and time ranges corresponding to them.

1 Based on the RNN frame, we accomplished the model construction, formula derivation and algorithm testing for PNN. We elucidated the mechanism of PNN based on the latest MIT research on synaptic compensation, and also grounded our study on the basis of findings of the Stanford research, which suggested that synapse formation is important for competition in dendrite morphogenesis. The influence of astrocytic impacts on brain plasticity and synapse formation is an important mechanism of our Neural Network at critical period or the closure of critical period. In addition to formula (1) and (2), which is derived from Back Propagation and features the synaptic connection weight and the range of action weight respectively. We also managed to figure out synaptic competition by current and mnemonic gradient informational synapse formation at critical period through formula (2) , synaptic compensation through formula (3), and current and mnemonic brain plasticity at critical period through formula (4). 2 The mnemonic gradient informational synapse formation needs to consider the forgotten memory-astrocytic synapse formation memory persistence factor. Mnemonic brain plasticity involves plus or minus disturbance-astrocytic brain plasticity phagocytose factor. The influence of astrocyte made local synaptic action range remain in an appropriate length at critical period. We give decreasing weights of factors at critical period. We try to deduce the mechanism of failure in brain plasticity by model at the closure of critical period in details by contrasting with previous studies. The simulation in Tables similar to that failure to the closure of critical period results in neurodevelopmental disorders [18] .The findings of our study proved meaningful, which revealed that much like astrocytic phagocytose factor, memory persistence factor also inhibit local synaptic accumulation [20] . The memory persistence factor obtained by the simulation of the mathematical model also inhibits the local accumulation of synapses. Furthermore, with the help of simulation, is it possible to obtain the same results through biological experiments to reduce the number of animal experiments and their suffering? 3 Test results of the 3 scenarios all concern mnemonic optical solution. All the PNNs with gradient-optimized synaptic action range fared better than those with random synaptic action range, which in turn outperformed PNNs with constant synaptic action range. Constant synaptic action range is achieved when the minimum synaptic action range grows to coincide with the maximum synaptic action range, which usually takes place in the cognition of the senile phase. The iteration of CRPNN is difficult to convergence due to a lack of plasticity and diversity, and RRPNN demonstrates slow convergence in testing. 4 We believe our work is of profound significance for the development of NN research, as we not only took the connection weights of synapses into consideration, but also paid heed to the current and mnemonic range of action weights-brain plasticity. Furthermore, PNN takes into account the mnemonic gradient informational synapse formation, and brain plasticity and synapse formation change frame of NN is a new method of Deep Learning. We can obtain better results in Tables of correlation coefficients respectively, when ORPNN contains both astrocytic synapse formation memory persistence factor and astrocytic brain plasticity phagocytose factor at critical period. 5 We developed the concept of weights for NN and divided the weights into categories, namely weights of synaptic connection and range of action. The latter reflects synaptic competition, compensation, current and mnemonic brain plasticity and gradient informational synapse formation at critical period. From input to output units, they go through hidden layers, and when the lead synaptic action range changes based on current and mnemonic brain plasticity and gradient informational synapse formation at critical period, these neurons, like a school of fish, also affect the synaptic position of the posterior neurons. 6 We also attempted to simulate PNN cognition process from infancy to the senile phase, and by analyzing relationship between different synapse formation and correlation coefficients. We may reasonably assume that the minimum synaptic action range increases and synaptic population decreases for synapse formation. 7 In the next step of our study, we took the real pandemic data into consideration, and PNN may well be used to process the pandemic data of the original strain and the variant Delta strain, in an effort to optimize the time interval of the synaptic action range, and then different connection weights will be applied for cure and infection rates. We will calculate the connection weights, and the synaptic action range, based on which we then formulate contingency plans for different period. 8 By testing, we made a similar observation to that of Dr. Luo's team through PNN: synaptic formation to a certain extent is detrimental to dendrites, and synaptic growth leads to a reduction in changes of the synaptic action range, therefore disrupting diversity and plasticity, including results considering memory persistence factor and phagocytose factor separately. 9 The synaptic competition mechanism NN is suitable for RNN frame as well as BP, and our next step is to conduct research on the implementation of synaptic competition, compensation and critical period in BP. 10 In simulation, if a same type of connection weight changes, all the range of action weights may be consequently updated for better results. 11 To be honest, the influence of astrocyte impacts on brain plasticity and synapse formation was inspired by Particle Swarm Optimization (PSO) [31] and consolidation of concrete, and critical period of brain plasticity really depends on inertia weight of PSO [29] . Only synapse formation impacts on brain cognition refers biological tests of neuroscience [15] , but also is relevant to the FEM. Synaptic compensation is actually inspired by the conservation of energy. The question we proposed is whether the promotion of neuroscience and brain cognition was achieved by model construction, formula derivation or algorithm testing, and whether we can reduce the number of animal experiments and their suffering through the guidance of model simulation planning. We resorted to the Artificial Neural Network (ANN), Evolutionary Computation and other numerical methods for hypotheses, possible explanations and rules, rather than only biological tests. 

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The authors declare no competing interest