A Non-linear Synaptic Neural Network Based on Excitation and Inhibition

Abstract

Based on the probabilistic features of synaptic excitation and inhibition, we propose and analyze a synaptic neural network. This synaptic neural network consists of synapses and neurons as well as their connections with synaptic graphs and neuronal networks. It simulates some of the computational properties of biological neurons and synapses. We define a non-linear synapse as a function of the excitatory probability of alpha-channel and the inhibitory probability of beta-channel. As a showcase, we create examples of synaptic neural networks. They're single synapse, dual synapse, and many synapse neural networks. In particular, we have concluded Boolean logic as a special case of synaptic neural networks. For practical applications, we provided and analyzed two solutions for synaptic neural networks to Content Addressable Memory (CAM) and Traveling Salesman Problem (TSP). In conclusion, synaptic neural networks can solve large-scale and difficult AI problems.

Full text

A NONLINEAR SYNAPTIC NEURAL NETWORK
BASED ON EXCITATION AND INHIBITION
CHANG LI
Abstract. A synaptic neural network based on the synaptic excitation and
inhibition is proposed and analyzed. This synaptic neural network consists of
synapses and neurons along with the connection of both the synaptic graphs
and the network of neurons which simulates some computational properties of
the biological neurons and synapses. A non-linear synapse is defined as one out-
put, one excitation input (α−channel), and one inhibition input (β−channel)
in which the output effect of a synapse is decided by the excitatory probability
and the inhibitory probability. Several special synaptic neural networks with
a single synapse, dual synapses, and many synapses are constructed. In par-
ticular, a Boolean Logic is treated as a simple case of a special synaptic neural
network. Finally, both Content Addressable Memory (CAM) and Traveling
Salesman Problem (TSP) applications are analyzed and illustrated. In conclu-
sion, the synaptic neural network is suitable for large scale and complicated
neural network applications.
1. Introduction
The research that simulates the biological neural network has made progress. Af-
ter the classic model of [McCulloch and Pitts (1946)], many neural network models
have been proposed, such as the nonlinear graded-response model and the percep-
tion model of the layer network.
These neural network models abstract some computational properties of the bi-
ological neural network. At the same time they illustrate the capability to solve
hard problems such as pattern recognition, associative memory and deep learning.
The synaptic neural network model proposed in this article is based on the proper-
ties of excitation and inhibition of the biological neurons and synapses. A synapse
consists of an excitatory input, an inhibitory input, and an output while a neu-
ron summons the inputs from synapses and other neurons to decide its activity.
A synaptic neural network is a non-linear dynamical system that is connected by
synapses and neurons. It may run into a stable state, an oscillate state, or a chaos
state. Even a simple network can demonstrate extremely complicated computa-
tional behavior.
Non-linearity, excitation and inhibition are fundamental behavior of the synaptic
Date: October 25, 2017.
Key words and phrases. synaptic neural network, neural network, nonlinear, neuron, synapse,
deep learning, memory, cam, optimization problem, tsp, probability, Boolean logic, excitation,
inhibition.
1

2 CHANG LI
neural network. Another characteristics are the non-determination and random-
ness of the synaptic neural network which can be understood by probability and
topology.
2. Non-linear Synapse and Probability
Suppose a synaptic neural network contains neurons in which they are connected
through non-linear synapses. A synapse consists of an input from the excitatory
α-channel, an input from the inhibitory β-channel, and an output channel which
sends a value to other synapses or neurons. Synapses may form a graph to accept
inputs from other neurons and output to a neuron. In advance, many synaptic
graphs can construct a neuron graph. Let’s start to describe a synapse.
Definition 1. Suppose S(x,y) is the non-linear function of a synapse from its
output channel,
(2.1) S(x, y) = αx(1 −βy)
where x∈(0.0,1.0) is the input variable from excitatory α-channel and α > 0.0
is the parameter of excitatory channel; y∈(0.0,1.0) is the input variable from
inhibitory β-channel and β > 0.0is the parameter of inhibitory channel.
Figure 1. A Non-linear Synapse S(x, y) = αx(1 −βy)
Now we are going to explain the equation (2.1) from the probability theory.
Considering the membrane of a synapse to which neurons are connected. There
are many channels (holes) on the surface of the membrane of a synapse where the
various kinds of ions can flow in and out. It is reasonable to assume below:
(i) The channels on membrane are divided into two types: one is called excita-
tory channel (α-channel), its opening increases the activity of the neuron; another
is called inhibitory channel (β-channel), its opening limits the activity of the neu-
ron. It is called channel selectivity.
(ii) An assumed variable of the neuron (i.e. conductivity rate) is related to the
number of opening channels. The more the opening excitatory channels, the bigger
the variable value; the more the opening inhibitory channels, the smaller the vari-
able value.

A NONLINEAR SYNAPTIC NEURAL NETWORK BASED ON EXCITATION AND INHIBITION3
If the opening probability of the excitatory channels at time tis x, and the opening
probability of inhibitory channels at time tis y, then the opening probability of
the connective synapse is x(1 −y). Suppose the linear relation of opening channel’s
probability and neuron variable (conductivity), the active probabilistic variable
value of a synapse can be represented as αx(1 −βy). The αis the excitatory pa-
rameter (α-channel) while the βis the inhibitory parameter (β-channel).
Figure 2. Synaptic Function S(x, y) = x(1 −y)
The biological research has demonstrated that the structure and the change of
the membrane protein is the key of the neuron activity. The change of the channel
protein on neuron and synaptic membrane (i.e. potential gate control channel and
chemical gate control channel shows the selectivity and threshold) can explain the
interaction among neurons and synapses. [David H. Hubel, Stevens C.F., Eric R.
Kandel, et,al 1985][3].
The Na+ channel illustrates the effect of an excitatory α-channel. The Na+ chan-
nel allows the Na+ ions flow in membrane and make the conductivity increase, then
produce excitatory post-synapse potential. The K+ channel illustrates the effect
of an inhibitory β-channel. The K+ channel that lets the K+ ions flow out of the
membrane shows the inhibition. This makes the control channel of potential gate
closing and generates inhibitory post-potential of the synapse.
Other kind of channels (i.e. Ca channel) have more complicated effects. Biological
experiments showed that there were only two types of channels in a synapse while
a neuron may be related to more types of channels on the membrane. Experiments
illustrated that while a neuron is firing, it will generate a series of spiking where
the spiking rate (frequency) reflects the strength of stimulation.

4 CHANG LI
In statistical physics, the relation between probability distribution Prand the en-
ergy Eris Pr=e−bErwhere the e−bEris called Boltzmann factor. Replacing x
and y in Eq.[2.1] by probability distribution, there is the formula αe−au(1−βe−bv ).
We can assign various probability distribution as the inputs of a synapse. There-
fore, a synaptic neural network can be considered as a dynamic probability network.
The stochastic attribute of channel’s opening and closing shows that the opening
time of special channels is random. The procedure that chemical material affects
the control channel of a chemical gate is completed by random procedure of the
mixing of the tokens of the small bulbs and membrane. A large number of channels
opening and closing shows the stochastic change. The probabilistic explanation of
neuron and synapse activity does make sense for biological neurons and synapses.
3. Synaptic Neural Network
The features of a synaptic neural network contain highly interconnected excita-
tory and inhibitory synapses that form a non-linear dynamical system. It is flexible
to construct a synaptic neural network with various topologies.
Definition 2. Suppose αki is the parameter of the non-linear excitatory synapse
between neuron xkand neuron xiwhile βkj is the parameter of the nonlinear in-
hibitory synapse between neuron xkand neuron xj. The input value of the excitatory
channel of the synapse is xkand the input value of the inhibitory channel of the
synapse is xj, the output to neuron xkis described below,
(3.1) xk=αkixi(1 −βk j xj)
The activation function of the neuron xkis
(3.2) xk={δ xk< ϵ (notfiring)
1−δ xk>1−ϵ(firing)
where αki, βkj >0; δ, ϵ ∈(0.0,1.0)
To input and output variables, the equation (3.2) is a threshold function. A
vector X= (x1, x2, ..., xn) can represent 2nlimited points by the firing and not
firing of output variables xk. If the vector is Xni= (xn1, xn2, ...) after the state
transformation, then the vector X is mapped to the vector Xni. All the vectors of
X that maps to the Xniconstructs an element in the classification set.
It is important to know that a nonlinear relation can act as an amplifier that
rapidly amplifies initial little variance to bigger in the final states. The equations
of synapses are the expression of the excitation and inhibition with nonlinear effects
in a synaptic neural network.
It is reasonable to assume that there are two classes of communication systems
for a synaptic neural network. A fired neuron outputs a spiking plus to synapses of
other neurons through axon while these synapses convert the spiking into energy
and complete a continuous non-linear computing then send to neurons which de-
cide their firing state. A synaptic neural network may have a loop connection that
constructs a continuous computing until some conditions satisfied.

A NONLINEAR SYNAPTIC NEURAL NETWORK BASED ON EXCITATION AND INHIBITION5
In our current model of synaptic neural network, the role of a neuron acts as an
oscillator to send out spiking pluses has been ignored. So a neuron simply input
and output a vale from an activation function.
3.1. Single Synapse. The Logistic Equation [3.3] has been deeply studied and
widely used in physics, ecology and chaos. [Peityen H.O., Richter P.H. (1986)][12]
Definition 3. The recurrence relationship of a single synapse is defined as
(3.3) x(n+1) =αx(n)(1 −βx(n))
If 0 ≤α≤3 the Eq.[3.3] has a single attractor, and it iterates to a stable state.
If 3 < α < 3.4444 there is a limit ring, the dynamical system has 2, 4, 8, ... finite
attractors. If α > 3.57, the chaos beginning, the series xnjumps from here to
there indeterminately, and the set of attractors have the attribute of fractals. One
Figure 3. Single Synapse in a Loop
chaos theory regards that the stretching and folding is the basic operation of chaos.
[Crutchfield J.P., Farmer J.D., Packard N.H., Shaw R.S. (1986)][2]. In Eq.[3.3] the
item αx(n)represents the stretching (excitation) operation, and the item 1 −βx(n)
represents the folding (inhibition) operation. This implies that chaos can exist in
the synaptic neural network.
3.2. Dual Synapses. Many types of connection for dual synapses exist. The sym-
metric loop connection below constructs a memory unit that can store the state of
this simple synaptic network.
Definition 4. Suppose α1, α2>1, β1, β2>0and x(0), y(0) ∈(0.0,1.0),
(3.4) {x(n+1) =α1x(n)(1 −β1y(n))
y(n+1) =α2y(n)(1 −β2x(n))
since x(n+1) −y(n+1) =α(x(n)−y(n)) if (α1=α2, β1=β2) and x(0) ≤y(0)
then x(n)≤y(n). If x(0) =y(0) , then the Eq.[3.4] is the same as the Eq.[3.3] If
α > 1, β = 1 then
(3.5) 




x(0) > y(0) →(1,0)
x(0) =y(0) →(0,0)
x(0) < y(0) →(0,1)
In the final state, the 1 represents the 1−ϵand the 0 represents ϵ, a very small real
number. The dynamical system of Eq.[3.4] constructs an oscillator with two stable

6 CHANG LI
states and its initial x, y value corresponds different stable state. If x(0) and y(0) is
not equal exactly, the synaptic neural network converges to (1, 0) or (0, 1) rapidly.
It completes a comparative operation and amplifies the small difference.
Figure 4. Dual Synapses as a Memory Unit
3.3. N Synapses. Let’s check the function S(x, y) = αx(1 −β y). If αis large, the
S(x, y) is large. But if βis large, the S(x, y) is small. In many constrain conditions
the inhibitory condition is easy to express. Generally, αcan be set as the same
value, the different inhibition can be expressed by β. One insight in the neural
network is the minimization principle which can be represented by different value
β.
Definition 5. Suppose there are m neurons and (m−1)msynapses, they can be
connected as a matrix of synapses below
(3.6) x(n+1)
i=αx(n)
i(1 −x(n)
1)(1 −x(n)
2)...(1 −x(n)
m)
i= 1..m, α ≥1.0,0.0< xi<1.0,x1, x2, ..., xminputs, αij =α1/(m−1), βij = 1.0.
Figure 5. m Neurons and (m−1)mSynapses.

A NONLINEAR SYNAPTIC NEURAL NETWORK BASED ON EXCITATION AND INHIBITION7
After iteration the neural network may arrived to the last stable state. The xi
has a value 1 if the xiis the maximum value. otherwise the xiis 0. Eq.[3.6] is a
m-stable state oscillator, it generates the maximum value among m variables like
a softmax function. This kind of synaptic neural network can be applied to build
CAM (Content Addressable Memory) and solve Optimization Problem.
4. Boolean Logic
Selecting suitable αand βin the synaptic neural network, we can express the
discrete DIFF (DIFFERENCE) in Boolean Logic. Actually when we set α=β= 1,
The function of the synapse becomes x(1 −y) which is the DIFF (DIFFERENCE)
logic xˆy. Since Boolean Logic is 1-bit binary logic with value 0 or 1, we can build
a 1-bit synaptic neural network.
Definition 6. The Boolean Logic D IF F, N OT, AN D, OR, X OR are represented
by synaptic equation below,
(4.1)















DI F F :xˆy =x(1 −y)
NOT :ˆx = 1 −x
AND :x&y=xy
OR :x|y= 1 −(1 −x)(1 −y)
XOR :x⊕y= 1 −(1 −x+xy)(1 −y+xy)
In binary state, x, y have value 0 or 1.
If the values of x and y are continuous, the non-linear synapse can be computed
in continuous variables. The Synaptic Logic Circuits can be implemented with
these basic Boolean synapses.
5. CAM (Content Addressable Memory)
CAM is an application of neural network. A CAM synaptic neural network forms
a parallel mapping from inputs to outputs. The information is stored as the weights
of the connection among synapses. Memorized data is recalled by their contents.
Adjust the local parameters and initial values might change the memorized content
by collective computation of network. Look at the connection of CAM, each neuron
connects all the other neurons by direct inhibition and it also forms a loop to itself.
Definition 7. The rule of transformation is
(5.1) xi=rdiixi(1 −dij xj)
xiis the output of a neuron or a synapse, the matrix dij is the link strength
between neuron i and j. dii is the excitatory parameter while dij is the inhibitory
parameter and r is the excitatory constant.
Definition 8. Selecting a matrix dij below, we can construct a CAM.
(5.2) dij ={∑(1 −Vi∗Vj)i̸=j
∑Vi∗Vji=jVi={ϵ xi= 0
1−ϵ xi= 1 ϵ > 0
Suppose U is the set of memorized word, the length of vector X= (x1, ..., xN) is
N, the number of memorized words in U is n. xiis equal to 0 or 1, the binary words
of X (i.e. 1000101 ...) represents the memorized information in the synaptic neural

8 CHANG LI
network. ϵis a very small positive real number which represents 0 of a continuous
variable and 1−ϵrepresents 1. If the bit i of memorized word is 1 then its Vi= 1−ϵ,
otherwise Vi=ϵ.
Since dij =dji , the matrix is a symmetric Hermit matrix which means that xi
and xjhave the same inhibitory strength. This selection of memorized matrix is
similar to other CAM models [Hopfield J.J. 1982][7]. Hermit matrix shows the sta-
bility of the neural network as a dynamical system.
Initial input of xiis a continuous variable, and the word of 0 and 1 is the last
stable state. If the corresponding input state is equal to the output state, then the
neural network memorizes the word, otherwise it corresponds another stable state.
This procedure is called recall of memory.
The vector length N=3, 50 and the memorized word length n=1,2,3 have been
selected for the numerical experiments. The result is summarized as below:
(i) Constant State. 000 ... is a special state of the synaptic neural network and
is always memorized in the synaptic neural network.
(ii) Complementary Memory. The complementary state of a memorized state is
always a correct answer. That is if 100 is memorized then 011 is memorized
as well.
(iii) Repeat Memory. Memory a lot of data information repeatedly does not change
the initial memory result.
(iv) Overload. To N=3, n=1,2, the r can be chosen that make all memory infor-
mation response correctly. When r≥3.0, no r can make all the response
correctly. For example, memory 100, 011 and 010, the 010 cannot response
correctly. It shows the number of content correlation is going to affect the
correctness of memory.
(v) Excitatory Parameter. The numerical experiments showed that the value of
r plays an important role on result. If r is large, the network responses the
memorized state with maximum strength, not all states can response correctly,
that is the sensibility is low and it cannot find the little difference. If r is too
small, the state without memorized vector can also be responded correctly,
that is the sensibility is too high.
When N=3, n=1,2, there exists r that all the memorized data responses correctly. If
n≥3, there are some errors. Because there are 2Ninformation can be memorized,
the maximum correct rate of all is 25%. When N is large the maximum correct
rate cannot be more than 25
In particular, N=50, n=6, and floating point real value ZERO = 0.00001, real
value ONE = 0.999, and the special memorized sequence is 0...00001, 0...00110,
0...01010, 0...10001, 0...10011, 0...11010. The experiment shows that the recalling

A NONLINEAR SYNAPTIC NEURAL NETWORK BASED ON EXCITATION AND INHIBITION9
strength which corresponds to the memorized value is (5,3,4,6,1,2). The smaller
the value, the bigger the strength to different r.
Suppose the correct rate = (correct response of memorized data number)/(memorized
data number). If r∈[1.1,1.5], all stored data can be recalled correctly, however
some not stored data can also be recalled correctly, the correct rate is 58%. If
r∈[1.8,2.0], the stored data cannot be recalled completely, but the correct rate is
80%. If r= 1.6749222 then the maximum correct rate is archived at 87.5%, no r
can be selected over that rate.
Remark 1. The rule of CAM model is similar to the Hebbian rule. The capability
of associative memory is in the correlation of the nonlinear synapses. Memory and
learning is related to the conductivity of neurons and synapses. A pair of neurons
in the different state will decrease the connective strength while a pair of neurons
in the same state will increase the connective strength.
In the CAM model, associative strength matrix is decided by inhibitory parame-
ters. Because the parameters are decided by the number of 0 and 1 in the sequence,
different place of the sequence is equal to value of excitatory parameter by 0 or 1.
Remark 2. Another model of CAM is constructed by layer network that is decided
by the content of memory. Like RNN, to a memorized sequence, different value of
bits is connected by inhibition channels, all inhibitory parameters are 1 while exci-
tatory parameters are r. Second data is memorized by the output of the first layer
as the input of the second layer, and so on. Finally, the last layer outputs value to
the first layer and forms the memorized network by feedback.
Although the experiment shows the capability of associative memory, it is absent
flexibility and regularity in somewhat. The memory implemented by the topological
connection can be applied to inference after the training of synaptic neural network.
Remark 3. In the previous CAM model, the parameter matrix is set in fixed.
However, we can change the strength according to the inputs of neurons by Hebbian
learning rule such that ∂dij =−∂Vi∂ Vj. Decreasing inhibitory connection strength
dij of different states, and increasing dij of the same state, that will increase the
reliability of memorized state. Since dij =dji , two connected neurons have the
same inhibitory parameter. Besides, the initial states of xican be set in random
values.
6. Optimization Problem
Optimization problem is the problem that searches for the maximum or mini-
mum under some constrain conditions. These problems are often in combinatorial
complexity. In the real world many optimization problems face the fact that to find
the best solution the exponential time has to be taken. Therefore the optimization
problems can be converted to find ’good’ solutions in polynomial time.
‘Traveling Salesman Problem’ (TSP) is a NP-complete problem that its features
and algorithms have been widely analyzed. [Garey M.R., Johnson D.S. (1979)[5],
Karp R.M. (1986))[10]]. Meanwhile, it is one of the most hard problem in the
NP-complete problems. Hopfield and Tank has presented a method that produces

10 CHANG LI
approximate solution with an analogy neural network, and showed the capability
to solve this problem rapidly. ([Hopfield J.J., Tank,D.W. (1985)][9]. Below I am
going to discuss how to solve the TSP by a synaptic neural network.
Definition 9. Suppose there are n cities A, B, .., AB, AC, ..., the distance from A
to B is dA,B, the Traveling Salesman Problem is to search a route that the salesman
visits each city once and only once, returns to the beginning city, and requires the
sum of the distance is minimum.
(6.1) xij =r4n−2xij
n
∏
k̸=i
(1 −xik)
n
∏
k̸=j
(1 −xkj )
n
∏
k=1
(1 −dk,ixkj⊖1)
n
∏
k=1
(1 −dk,ixkj⊕1)
(6.2) 1≤i, j, k ≤n, 0< r, 0< dij ≤1, dij =dji
(6.3) j⊖1 = {j−1j > 1
n j ≤1j⊕1 = {j+ 1 j < n
1j≥1
To exactly represent the problem, the matrix of n2is used. The row represents the
city name, the column represents the order which the cities are visited. The value
1 of item at (x, i) represents that the city x is visited at time i, otherwise value 0
means not visited at time i. The condition is translated to that there are only 1 in
each rows and columns. The total visited distance is dA,X1+dX1,X2+... +dXn,A
where Xiis the city that its i-th column is 1 and A is the beginning city.
Figure 6. Synaptic Neural Network for TSP (Partial Graphs)

A NONLINEAR SYNAPTIC NEURAL NETWORK BASED ON EXCITATION AND INHIBITION11
To each neuron the inhibitory connection of each row and column units keeps
only one neuron excitation in each row and column. First two items of product
in Eq.[6.1] makes these conditions possible. All the inhibitory parameters of the
synapses are set to 1.
The distance of path from city X to next city Y is dX,Y . To city X, the distance
of the next possible cities is inhibitory parameter, and it inhibits the connection of
X, so the shortest path has the largest probability to be selected. This method is
similar to the neighbor heuristic algorithm, and will increase the smallest inhibitory
items, that is the shortest distance. The city which connected before has the same
connected mode. All the neurons are connected so that the first column and the last
column is connected in a loop. Last two items of product in Eq.[6.1] implemented
the heuristic algorithm.
An example of n= 10 is selected to test from [Floulds I.R. (1984)][4]. It is in-
terested to find that the results changed by various r value so that there are the
best solutions, good solutions, and oscillation. Even if a little bit change of r was
made, it is possible to generate different distances and visiting sequence.
Although this TSP synaptic neural network does not always generate the best
solution, it is similar to a heuristic algorithm to get good by selecting value r. Ev-
ery synapse has to do 6n−2 multiplications and 4n−2 subtractions, and the total
of synapse is 2n2, therefore each iteration time is O(n3). Suppose the converge time
is O(n2), the total computing time is in O(n5).
There are some interesting features of this synaptic neural network for optimization
problem. Firstly, the algorithm of the synaptic neural network is non-deterministic
according to the value r. The assumed r can be selected and achieved to the best
solution in polynomial time. If N P ̸=Pthere are not exist an algorithm to all
instances so that the rate of the solution with the best solution RA < ∞[Garey
M.R., Johnson D.S. (1979)][5]. That is we cannot find a TSP algorithm so that its
every solution is very near the best solution. But the research has proved that the
non-deterministic algorithms have the capabilities to check an exponential number
of possibilities in polynomial time.
Secondly, the synaptic neural network is parallel in computing but it can be in
synchronization or non-synchronization. Here we approached the computing mode
in synchronization.
Finally, the nonlinear relation has the feature of exponential computing. Inves-
tigating the change of each synapses and neurons, we find that from beginning
although the difference is very small, the difference is amplified, at a stage of pro-
cessing, the change dramatically arrived to the stable state or in oscillation. This
non-linear computing capability of continuous variable is the key to solve the prob-
lem in polynomial time.

12 CHANG LI
7. Conclusion
A non-linear synaptic neural network based on excitation and inhibition has been
analyzed and simulated. In particular, Boolean Logic was concluded as a special
synaptic neural network. Two applications on both memory (CAM) and optimiza-
tion problem (TSP) were demonstrated. Like a transistor, a synapse acts as a basic
non-linear computing unit to construct the synaptic neural network. We also il-
lustrate that a synaptic graph itself can complete many complex computing tasks.
Moreover, the synaptic neural network provides a simple prototype for research of
brain with complex topological connection.
Multi-core CPU, Multi-thread GPU, Multi-cell FPGA are current suitable comput-
ing devices for the implementation of the synaptic neural network. The framework,
toolkit, and languages for synaptic neural network can be efficient components and
tools to build the ecosystem of the synaptic neural network for AI applications.
In the future we can apply the Synaptic Neural Network (SNN) on Deep Learning
(DL), Recurrent Neural Network (RNN)and Convolutional Neural Network (CNN)
construction for new generation AI applications. The implementation of the Synap-
tic Neural Network (SNN) by semiconductor, optical circuit, and/or chemistry re-
actor can solve many computing problems in ultra fast speed with big data and low
power consumption.
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A NONLINEAR SYNAPTIC NEURAL NETWORK BASED ON EXCITATION AND INHIBITION13
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E-mail address, Chang LI: changli@neatware.com
URL:https://www.researchgate.net/profile/Chang Li59/contributions