# Simulation of Spinal Cord Reflexes
## 2 Theoretical Considerations  
The neuron can be considered a circuit working with frequency modulated pulses. The main characteristic of the pulses is that the duty time of every pulse is the same no matter how long the entire pulse period is. As a consequence, the modulation addresses only the duration of the pulses' pause time. A diagram with the idealized pulse parameters is given in Figure 1. First of all, in order to consider a frequency modulation taking place, we must assume a central frequency, or a characteristic frequency, of every neuron type, modifiable according to a modulation function \(f()\). Considering an infinite pulse train for every pulse n, we have a value \(f(n)\) of the modulation function determining the pause timeof the \(n\) pulse, prolonging or protracting it according to its sign. Theoretically, the pulse train can be defined as follows:  
\[f:N\rightarrow(T-L,T+L),\ L\ <\ T-tp, \tag{1}\]  
where \(T\) is the pulse period at the neuron characteristic frequency. If \(tp\) is the duty time of the pulses, the Laplace transform of the pulse train is:  
\[\begin{array}{ll}I(s)&=\int_{0}^{\infty}e^{-st}f(t)dt=\sum_{n=0}^{\infty} \int_{0}^{tp}He^{-st}dt-\sum_{n=0}^{\infty}\int_{T}^{T}he^{-st}dt-\sum_{n=0}^{ \infty}\int_{0}^{f(n)}he^{-st}dt\\ &=\sum_{n=0}^{\infty}\left(\int_{0}^{tp}He^{-st}dt+\int_{tp}^{T}he^{-st}dt \right)+\sum_{n=0}^{\infty}\int_{0}^{f(n)}he^{-st}dt=\textit{If}(s)+\textit{lv} (s),\end{array} \tag{2}\]  
where _If_(s) is the fixed part and _lv_(_s_) is the variable part of _l_(_s_). The constant part is the integral of a periodic function, _fp_(_), respectively, of a pulse train with the characteristics shown in Figure 1. The integral part of a periodic function is given in textbooks, and _If_(_s_) can be written as:  
\[\textit{If}(s)=\int_{0}^{T}e^{-st}f_{p}(t)dt\Bigg{/}(1-e^{-sT}) \tag{3}\]  
\[f_{p}(t)=\left\{\begin{array}{ll}H;\ t\in\left(0,t_{p}\right)\\ -h;\ t\in\left(t_{p},\ T\right)\end{array}\right. \tag{4}\]  
\[\textit{If}(s)=\frac{1}{s(1-e^{-st})}(H(1-e^{-st_{p}})+h(e^{-sT}-e^{-st_{p}})) \tag{5}\]  
\[\textit{lv}(s)=\sum_{n=0}^{\infty}\frac{h}{s}\Big{(}1-e^{-sf(n)}\Big{)} \tag{6}\]  
The part of the signal that is of interest to us is the variable one. As can be seen, the modulation part of the signal, _lv_, is already separated from the carrier _If_. The final separation of the modulation from the pulses is achieved at the effectors. Overall, we consider the neuron to be a system with variable parameters [4]; yet, during a certain time interval, it can be approximated as a linear system with fixed parameters. A neuron needs a certain number of excitation pulses, always the same, coming at a convenient time interval, to give a single pulse at the output [1]. The number of excitation pulses can be obtained by time summation and/or by spatial summation. Our neuronal scheme models take this fact into account. The ratio between the output pulses and the input excitation pulses does not remain constant. When the frequency of the excitation pulse input rises, approaching the limit of 100 Hz, and the pause time approaches the refractory period, the ratio obviously decreases. The same phenomenon occurs in our simulations due to the capacity of the charge/discharge.