## 3 Discussion
A membrane potential of about \(V_{m}=100\) mV and a soma radius of about \(R=25\)\(\upmu\)m result in an ejected charge \(Q=8\pi\epsilon_{0}RV_{m}=5.55\cdot 10^{-16}\)\(C\). If the ejected charge \(Q\) crosses the axon in about 1 ms, the current impulse \(I(t)\) can be described by a Gaussian function of time with variance \(\sigma=1\) ms:  
\[I(t)=\frac{Q}{\sigma\sqrt{2\pi}}\exp\biggl{(}\frac{t^{2}}{2\sigma^{2}}\biggr{)} \tag{20}\]with the condition \(\int_{-\infty}^{\infty}I(t)dt=Q\). The current \(I(t)\) can be introduced in the expression of the magnetic field (Equation (18)), i.e.,  
\[B(t)=\frac{\mu_{0}}{4\pi}\frac{Q}{\sigma\sqrt{2\pi}}\exp\left(\frac{t^{2}}{2 \sigma^{2}}\right)\Delta\ell\frac{\sin(\theta)}{r_{s}^{2}}\ . \tag{21}\]  
The maximum value of the magnetic field amplitude is achieved at \(t=0\) with \(\sin\theta=1\). This situation, where \(\sin(\theta)=1\), occurs when the axon of the firing neuron is tangent to the surface of the skull in the position of the magnetic sensor. In the cortical area of the brain, this may happen when the firing neuron is in one of the numerous regions called gyri [28; 29]. By assuming an average value of \(\Delta\ell=100\)\(\upmu\)m and an average value of the distance of the magnetic sensor detecting the impulse of \(r_{s}=2\cdot 10^{-2}\) m, an average peak intensity \(B(0)=0.5\cdot 10^{-20}\)\(T\) is obtained. To generate magnetic impulses of the order of \(10^{-13}\)\(T\), a value generally found in MEG and EEG, about \(N_{i}=10^{7}\) synchronized firing neurons would be needed (the value \(10^{-13}\)\(T\) is close to the sensitivity limit of the SQUID sensors (about \(10^{-15}\)\(T\))). While it is generally accepted that the magnetic signal generated by firing neurons (MEG) is due to the ejected excess charge within the axon, the origin of the electric signal (EEG) is more controversial. A common assumption is that the signal is generated in correspondence of the chemical synapses connecting the firing neuron to the dendrites of numerous postsynaptic ones. The charge emitted from the axon of the firing neuron is split in hundreds or thousands of fractions which generate ionic currents external to the synapses, and are then detected by the sensors of the EEG setup. The synchronized firing of a large population of neurons and the role played by the excitatory and inhibitory synapses yield a complex situation of small current impulses, where spontaneous oscillations can be generated under suitable assumptions [30; 31]. An alternative interpretation relates the electric signals to the transient potential impulses radially emitted externally to the soma of the firing neurons, due to the rapid variations in the external electric field generated by the charge variations inside the neuron. Within this description, the potential impulse peak value, detected by an electrode internal to the soma during the firing event, is assumed of the order of 100 mV, the electrode external to the soma peak value of about 0.1 mV, and the distance of the second electrode from the membrane of the neuron of the order of 200 \(\upmu\)m. In order to evaluate the intensity of the electric potential impulse, we consider again the neuron with the spherical soma of radius \(R=25\)\(\upmu\)m and the electric field \(E(r)\) at \(r<R\) used above for the calculation of the magnetic impulse. If the conductive medium within the cranial bone is neglected, the electric field at \(r>R\) is given by \(E(r)=Q/4\pi\epsilon_{0}r^{2}\) where \(Q\) is the whole charge within the soma. The electric potential, at distance \(r>R\) from the center of the sphere, is given by \(V(r)=Q/4\pi\epsilon_{0}r\), with \(V=0\) as \(r=\infty\). The field \(E(r)\) in the presence of conductive liquids, as it is the case for the brain, in stationary condition would be zero. A charge layer equal to the internal charge but with the opposite sign, would be attracted and surround the membrane of the soma. This charge with spherical symmetry cancels out the field for \(r>R\).