The two subcritical Hopf bifurcations that have been found in this study have a precise neurobiological meaning. For each of the two bifurcations there is a stable fixed point surrounded by an unstable periodic orbit. Instability in this case means that if one looks for the periodic orbit in the

$(t,E)$ space, any small departure from it will bring the system of CA3 neurons out of this periodic orbit. An unstable limit cycle (similar to an unstable node or focus) means that the dynamic system intrinsically departs from it and falls either in a stable node or a stable limit cycle. Thus it is very difficult to reveal it during the simulation. When the parameter

$C{B}_{exo}$ approaches the critical value, which is in the interval

$(1.6,1.7)$ for the first transition and in the interval

$(1.8,1.9)$ for the second transition, the unstable periodic orbit collapses on the stable point which becomes unstable. The main effect is the change of stability of the equilibrium point of the neuron population.So for every value of the main parameter

$C{B}_{exo}$ ata the left of the first

H point in

Figure 1 there is an unstable periodic orbit, and another unstable periodic orbit appears after the second

H point. These cycles are important but rather difficult to catch numerically. The two Neutral Saddle Points (NSP) in the figure are two fixed points in the limiting case when the two eigenvalues are equal. The saddle point is a point which is stable in one eigenvalue direction and unstable in the other one, meaning that one eigenvalue is positive and the other is negative. So a saddle point is an unstable equilibrium point: in the special NSP case the two eigenvalues have equal absolute value. This means that a small perturbation of this motion in the stable direction should amplify the motion in a similar way as the perturbation in the other direction. The saddle-node bifurcation of the cycles (or limit point of cycles - LPC) is the analogous of the saddle-node bifurcation of equilibrium points. An unstable orbit collides with a stable one and they annihilate each other: in the case of equilibrium points a stable one approaches an unstable one and disappears. In terms of neuronal oscillations this means that there are two oscillations of the CA3 excitatory neurons, one stable and another one unstable, which approach each other and disappear for the values of

$(C{B}_{exo},E)$ shown in the picture. There is also a line of stable limit cycles connecting the two LPC cycles. These are stable oscillations of the

E population of neurons. The set of stable cycles corresponding to the green line of

Figure 3 is shown in

Figure 5,

Figure 6 and

Figure 7: the frequency changes in agreement with

Figure 3. In general these geometrical figures have a basin of attraction, so not all the LPC, NSP, unstable or stable orbits have been found during this study but the knowledge of their existence gives an indication for what values of parameters one can have such a behavior of the system. In order to see all of them it is necessary to choose in a proper way the initial conditions, which represents a lengthy and tedious work. We also underline that the behavior with respect to the

$C{B}_{exo}$ parameter is not changed if one varies the

b parameter of the endocannabinoids or the

$\beta $ parameters of the sigmoid function used in the main evolution Equation (

1). The other type of Hopf bifurcation is the supercritical one, which does not appear for this model. From the theory of stability of solutions of systems of differential equations it is well known that the sign of the Lyapunov coefficient determines the type of Hopf bifurcation. In our case these coefficients are both positive and so the transition is subcritical. The concept of bistability is explained in the section of the

Results. We want to mention here that it is clear from

Figure 4 that there is a block of the spiking activity of the

E population, and from

Figure 5,

Figure 6 and

Figure 7 that when

$C{B}_{exo}$ approaches the first Hopf bifurcation point the spiking frequency increases (

$1.6$,

$1.8$ and

$1.9$ are the values of the

$C{B}_{exo}$ parameter) and then for

$C{B}_{exo}=2$ the spikes vanish. In these figures

$C{B}_{endo}$ is reported and one can see that it oscillates with the activity of the group of the excitatory population of

$CA3$ neurons. This implies the oscillation of the synaptic coupling weights and so there is a learning process going on, which ends when the oscillations cease. The meaning of the phase diagram obtained by varying two parameters is more subtle but it can be analyzed in an analogous way. The model we used is a simplified mathematical model, it is intrinsically limited and cannot encompass the complexity and diversity of the effect of exogenous and endogenous endocannabinoids in vivo [

16]. The main effect of endocannabinoids on synaptic plasticity is long-term depression (LTD) (not long-term potentiation), first observed in the striatum [

17] and then in other brain structures [

1]. Furthermore smoking cannabis impairs short-term memory [

18]. Mice treated with tetrahydrocannabinol (THC) show suppression of long-term potentiation in the hippocampus [

19].