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Hebbian plasticity precisely describes how synapses increase their synaptic strengths according to the correlated activities between two neurons; however, it fails to explain how these activities dilute the strength of the same synapses. Recent literature has proposed spike-timing-dependent plasticity and short-term plasticity on multiple dynamic stochastic synapses that can control synaptic excitation and remove many user-defined constraints. Under this hypothesis, a network model was implemented giving more computational power to receptors, and the behavior at a synapse was defined by the collective dynamic activities of stochastic receptors. An experiment was conducted to analyze can spike-timing-dependent plasticity interplay with short-term plasticity to balance the excitation of the Hebbian neurons without weight constraints? If so what underline mechanisms help neurons to maintain such excitation in computational environment? According to our results both plasticity mechanisms work together to balance the excitation of the neural network as our neurons stabilized its weights for Poisson inputs with mean firing rates from 10 Hz to 40 Hz. The behavior generated by the two neurons was similar to the behavior discussed under synaptic redistribution, so that synaptic weights were stabilized while there was a continuous increase of presynaptic probability of release and higher turnover rate of postsynaptic receptors.

Even though Hebbian synaptic plasticity is a powerful concept which explains how the correlated activity between presynaptic and postsynaptic neurons increases the synaptic strength, its value has been diminished as a learning postulate because it does not provide enough explanation how synaptic weakening occurs. In simple mathematical interpretation of Hebbian learning algorithm, an increase of the synaptic strength between two neurons can be seen if their activity is correlated otherwise it is decreased [

However, the specifics of a biologically plausible model of plasticity that can account for the observed synaptic patterns have remained elusive. To get biologically plausible model and remove the instability in Hebbian plasticity many mechanisms have been discussed in recent findings. One remarkable suggestion is to combine STDP with multiple dynamic and stochastic synaptic connections which enable the neurons to contact each other simultaneously through multiple synaptic communication pathways that are highly sensitive to the dynamic updates and stochastically adjust their states according to the activity history. Furthermore, strength of these individual connections between neurons is necessarily a function of the number of synaptic contacts, the probability of neurotransmitter release, and postsynaptic depolarization [

A fully connected neural network was developed with two neurons in which each neuron consisted of thousands of computational units. These computational units were categorized as transmitters and receptors according to the role they played on the network. A unit was called a transmitter if it transmitted signals to other neurons and a unit was called a receptor if it received the signals into the neuron. The receptors of a given neuron were clustered into receptor groups. According to the excitation and the inhibition of the model neuron these computational units could update their states dynamically from active state to inactive state or vice versa. Only when a computational unit was in active state it could successfully transmit signals between neurons. Transmitters from presynaptic neuron and receptors of the corresponding receptor group of the postsynaptic neuron together simulated the process of a single synapse. Transmitter at presynaptic neuron can be considered as a synaptic vesicle which can release only a single neurotransmitter at a time and the model receptors can be considered as postsynaptic receptors at synaptic cleft. With these features, excitation of a neuron at a particular synapse in our network was determined by the function of the number of active transmitters in the presynaptic neuron, transmitters’ release probability, and the number of active receptors at the corresponding receptor group of the postsynaptic neuron. First, in order to analyze how network with two neurons could balance the excitation when Poisson inputs with mean rates 10 Hz and 40 Hz were applied, Only one neuron was fed by Poisson inputs while letting the other neuron to adjust itself according to the presynaptic fluctuations. Neurons stabilized its weight for both Poisson inputs while the weight values of Poisson inputs with mean rate 10 Hz were stabilized into higher range compared to when Poisson inputs with mean rate 40 Hz was applied. The analysis into internal dynamics of neurons shows that neurons have behaved similar to the process discussed in synaptic redistribution when long-term plasticity interacts with short-term depression. Further, neurons have played complementary roles to maintain the network’s excitation in an operational level. These compensatory roles have not damaged the network biological plausibility as we could see that neurons worked as integrators that integrate higher synaptic weighted inputs to lower output and vice versa. Finally the network behavior was evaluated for other Poisson inputs with mean rates in the range of 10 Hz to 40 Hz and observed as the mean rate of the Poisson inputs increases, the immediate postsynaptic neuron increases its synaptic weights, while the immediate presynaptic neuron of those inputs was settle, into a complementary state to the immediate postsynaptic neuron.

A fully connected network with two neurons was created. Each neuron was attached to thousands of computational units which were either in active state or inactive state according to the excitation and the inhibition of the attached neuron. Units attached to a neuron were classified into two groups based on the role they played on the neuron. A computational unit that transmitted the signal from the attached neurons to other neurons was called a transmitter and a computational unit that received the signals to the attached neurons from other neurons was called a receptor. Further, receptors attached to a neuron were clustered into groups so that transmitters from presynaptic neuron could contact the postsynaptic neuron simultaneously through multiple synaptic connections. Figure

Structure of neuron

The transmitters from presynaptic neurons contacted the receptors of a particular receptor group of postsynaptic neurons by forming a synapse between the two neurons; see Figure

Structure of the neural network.

When defining the process under dynamic stochastic synapses we have only concerned with the properties and mechanisms of use-dependent plasticity from the few milliseconds to several minutes time scales. Therefore, use-dependent activity to our modeled network was introduced using short-term plasticity; facilitation and depletion [

If

A modeled neuron maintained threshold values

Moreover, according to the following predefined behavioral rule signal was propagated between neurons.

Once a received signal is applied to a receptor if the receptor is updated to inactive state then the received signal is inactivated otherwise the signal is propagated to a randomly selected transmitter of the same neuron.

Once a transmitter of a particular neuron receives a signal at time step

The above behavioral rule defines the underlying mechanism of signal transmission between the presynaptic neuron and the postsynaptic neuron; that is, when the related computational units from the two neurons are active only, the signal is successfully transmitted. Therefore, the number of active receptors in a receptor group of the postsynaptic neuron and the number of active transmitters in the presynaptic neuron jointly define the efficacy at a given synapse. In addition to this short-term plasticity and homeostatic synaptic plasticity [

The process at synapses where transmitters from the presynaptic neuron contacted the receptors in a particular receptor group of the postsynaptic neuron were binned to analyze the synapse’s excitation. Bin is an array of seven columns,

This allowed us to define the time represented by each cell in a bin from its first cell as in (

Bin the process at a single synapse.

Let

The mean release probability of the presynaptic transmitters within a given bin, say

STDP is a form of long-term modification to synaptic strength that depends on the action potential arriving timing between presynaptic neuron

Here

Learning based on STDP was implemented on synapses assuming that bins of a given synapse are mutually independent and the impact that each bin made on the synapse sums linearly. Then mean

According to the model proposed in [

If we combined (

An experiment was arranged to find the possibility that can STDP and short-term plasticity together balance the excitation of a network with two Hebbian neurons without defining any constraints on the weight learning algorithm. A fully connected network with two neurons, say neuron

Network structure with ten synaptic connections. (a) shows the developed fully connected network to test how neurons could balance the excitation after external input was applied to part of it.

Figures

Distribution of the weights and release probabilities of neurons at Poisson inputs with mean rate 10 Hz. Each subfigure in the figure depicts the distribution of the weight algorithm and the mean release probability at the given synapse of postsynaptic neuron

Distribution of the weights and release probabilities of neurons at Poisson inputs with mean rate 40 Hz. Each subfigure in the figure depicts the distribution of the weight algorithm and the mean release probability at the given synapse of postsynaptic neuron

Distribution of the medians of synaptic weights at each synapse at Poisson inputs. The figure shows the variation of the median of the weight distribution at each Poisson input;

Next we were interested to know what makes the neuron to stabilize its activity without being overexcited or overdepressed in a network which has no controlling constraints. To understand that we analyzed the internal behaviors of neurons

Distribution of the coefficient of variation (CV) at each synapse at Poisson inputs with 10 Hz. Each subfigure Figure

Distribution of the CV at each synapse at Poisson inputs with mean firing rate 40 Hz. Each subfigure in the figure depicts the distribution of the CV at the given synapse of both neuron

Moreover, if the difference of the value of CVs between two neurons was considered, it is clearly shown in the Figures

Finally we would like to understand the behavior of the network when applying Poisson inputs in the range of 10 Hz and 40 Hz. The Poisson inputs with mean rates, 15 Hz, 20 Hz, 25 Hz, 30 Hz, and 35 Hz were also presented to neuron

Distribution of the median of synaptic weights of neurons at different mean firing rates. Figure

As per the literature, a synapse can be strengthened either by increasing the probability of transmitter release presynaptically or by increasing the number of active receptors postsynaptically. This general functionality at the synapses can be varied by the interplay between long-term plasticity and short-term dynamics, especially short-term depression. Short-term depression is mainly based on vesicle depletion which is the use-dependent reduction of neurotransmitter release in the readily releasable pool [

STDP has successfully interplayed with short-term plasticity to control the excitation or inhibition of neural network according to external adjustments. Notably, these adjustments are consistent and are also biologically plausible. The stabilization of synaptic weights in an operational level without controlling constraints seems to be possible if STDP as long-term plasticity interacts with the short-term dynamics. The dynamic behavior of short-term activity is necessary to propagate and balance the excitation of neural network without damaging the synaptic weight distribution; similar to how CV and probability of release have played with STDP to balance the excitation. When compared to Luz and Shamir [

The model proposed in this research is a computational model to investigate the internal dynamics of neural networks when STDP, Hebbian plasticity and short-term plasticity, are interacting with each other. The model itself has few drawbacks; mainly the neural network of our model has spent around 150 bins to show the adjustment to external modifications; this is mainly because we selected the median of the weight distribution as the amount of synaptic potentiated as a response to the pair of presynaptic and postsynaptic spike (in (