**Leftward shift case:** This case corresponds to a class of solutions where $\forall \beta \in E, \forall z \in Z^\beta, \psi_u^\beta(z+a) < 0$ and $\psi_u^\beta(z-a) > 0$. With the simplified equation, since all $\psi_u^\beta(z+a)$ (respectively $\psi_u^\beta(z-a)$) have the same sign, all $\psi_u^\beta(z+a)$ ($\psi_u^\beta(z-a)$) are equal.

The resulting equation is

$$ \left[ \Gamma_{\beta_q}(\lambda) - \pi_u(\lambda + \alpha_q^{-1})2w(0) \right] \left[ \Gamma_{\beta_q}(\lambda) - \pi_u/\xi_s(\lambda + \alpha_q^{-1})2w(0) \right] - 1/\xi_s \left[ \pi_u(\lambda + \alpha_q^{-1})2w(2a) \right]^2 + \frac{\lambda\alpha_q\beta_q/\xi_s\gamma_u}{1 + \alpha_q\beta_q} \left[ \left[ \Gamma_{\beta_q}(\lambda) - \pi_u(\lambda + \alpha_q^{-1})2w(0) \right]2w(0) - \left[ \pi_u(\lambda + \alpha_q^{-1})2w(2a) \right]2w(2a) \right] = 0, \quad (A.29) $$

with

$$ \Gamma_{\beta_q}(\lambda) = (\lambda + \alpha_q^{-1} + \beta_q)(\lambda + 1) \quad (A.30) $$

The roots of this equation can be searched for numerically, and must satisfy the conditions $\psi_u^\beta(z+a) < 0$ and $\psi_u^\beta(z-a) > 0$. But, since we do not compute them for any set of parameters, we cannot conclude on the effect of learning weights on stability in this case.

**Rightward shift case:** This case corresponds to a class of solutions where $\forall \beta \in E, \forall z \in Z^\beta, \psi_u^\beta(z+a) > 0$ and $\psi_u^\beta(z-a) < 0$. Due to the symmetry of the system, the spectrum associated with rightward shifts is identical to that of leftward shifts.

As a conclusion, even if we do not compute the eigenvalues with a given set of parameters, it was still possible to estimate the stability behaviours, provided that equilibrium bump solutions exist. As in the previous subsubsection Appendix A.1.1, we found that learning weights stabilize the system through contraction, but have no effects on stability in case of an expansion perturbation. However, we cannot conclude on their influence on the shift behaviour since we did not compute eigenvalues values.

All in all, this analysis provided an analytical framework to study the three neural fields model with spike frequency adaptation, synaptic depression and learning weights.
**Leftward shift case:**. This case corresponds to a class of solutions where $\forall \beta \in E, \forall z \in Z^\beta, \psi_u^\beta(z+a) < 0$ and $\psi_u^\beta(z-a) > 0$. With the simplified equation, since all $\psi_u^\beta(z+a)$ (respectively $\psi_u^\beta(z-a)$) have the same sign, all $\psi_u^\beta(z+a)$ ($\psi_u^\beta(z-a)$) are equal.

The resulting equation is

$$ \left[ \Gamma_{\beta_q}(\lambda) - \pi_u(\lambda + \alpha_q^{-1})2w(0) \right] \left[ \Gamma_{\beta_q}(\lambda) - \pi_u/\xi_s(\lambda + \alpha_q^{-1})2w(0) \right] - 1/\xi_s \left[ \pi_u(\lambda + \alpha_q^{-1})2w(2a) \right]^2 + \frac{\lambda\alpha_q\beta_q/\xi_s\gamma_u}{1 + \alpha_q\beta_q} \left[ \left[ \Gamma_{\beta_q}(\lambda) - \pi_u(\lambda + \alpha_q^{-1})2w(0) \right]2w(0) - \left[ \pi_u(\lambda + \alpha_q^{-1})2w(2a) \right]2w(2a) \right] = 0, \quad (A.29) $$

with

$$ \Gamma_{\beta_q}(\lambda) = (\lambda + \alpha_q^{-1} + \beta_q)(\lambda + 1) \quad (A.30) $$

The roots of this equation can be searched for numerically, and must satisfy the conditions $\psi_u^\beta(z+a) < 0$ and $\psi_u^\beta(z-a) > 0$. But, since we do not compute them for any set of parameters, we cannot conclude on the effect of learning weights on stability in this case.

**Rightward shift case:**. This case corresponds to a class of solutions where $\forall \beta \in E, \forall z \in Z^\beta, \psi_u^\beta(z+a) > 0$ and $\psi_u^\beta(z-a) < 0$. Due to the symmetry of the system, the spectrum associated with rightward shifts is identical to that of leftward shifts.

As a conclusion, even if we do not compute the eigenvalues with a given set of parameters, it was still possible to estimate the stability behaviours, provided that equilibrium bump solutions exist. As in the previous subsubsection Appendix A.1.1, we found that learning weights stabilize the system through contraction, but have no effects on stability in case of an expansion perturbation. However, we cannot conclude on their influence on the shift behaviour since we did not compute eigenvalues values.

All in all, this analysis provided an analytical framework to study the three neural fields model with spike frequency adaptation, synaptic depression and learning weights.
Appendix B. Supplementary figures

<figure><img src="image_11.png" /><figcaption>Figure B.1: The encoding phase shows faster learning in hippocampal fields than in neocortical ones. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). The initial stimuli are two bumps around positions $A^C$ and $B^C$ in the neocortical field. They activate C neurons which in turn activate D neurons. Then since C and D neurons fire, H neurons can be activated. In the end, neurons shutdown, due to the depletion of synaptic resources. (b) Intra and (c) inter-field learning weights at the end of the encoding phase. Learning weights within the $A^\alpha$ part of each pattern are at their maximum. Indeed, neurons inside this location are very close to each other, so their weights grow fast. On the opposite the weights between the $A^\alpha$ and $B^\alpha$ parts are smaller, since the distance is more important. Especially these cross weights are still smaller in C than in D and H fields, since the distance between $A^C$ and $B^C$ is larger than those between $A^D$ and $B^D$ (or $A^H$ and $B^H$). Thus in our model, distance is the main reason why the neocortex is a slow learner and the hippocampus a fast learner.</figcaption></figure>
Appendix B. Supplementary figures

<figure><img src="image_11.png" /><figcaption>Figure B.1: The encoding phase shows faster learning in hippocampal fields than in neocortical ones. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). The initial stimuli are two bumps around positions $A^C$ and $B^C$ in the neocortical field. They activate C neurons which in turn activate D neurons. Then since C and D neurons fire, H neurons can be activated. In the end, neurons shutdown, due to the depletion of synaptic resources. (b) Intra and (c) inter-field learning weights at the end of the encoding phase. Learning weights within the $A^\alpha$ part of each pattern are at their maximum. Indeed, neurons inside this location are very close to each other, so their weights grow fast. On the opposite the weights between the $A^\alpha$ and $B^\alpha$ parts are smaller, since the distance is more important. Especially these cross weights are still smaller in C than in D and H fields, since the distance between $A^C$ and $B^C$ is larger than those between $A^D$ and $B^D$ (or $A^H$ and $B^H$). Thus in our model, distance is the main reason why the neocortex is a slow learner and the hippocampus a fast learner.</figcaption></figure>
<figure><img src="image_12.png" /></figure>

Figure B.2: The hippocampal replay step allows engrams reactivation. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). We modeled hippocampal reactivation by sending a spatially uniform stimulus on H. All H neurons receive this input, but only those involved in the pattern fire for a long time, because of their non zero learning weights. Activated H neurons lead to firing in C neurons, which then activate D neurons. (b) Intra and (c) inter-field learning weights at the end of the hippocampal replay phase (solid lines) compared to learning weights at the end of the encoding phase (dotted lines). Weights consolidate thanks to pattern reactivation, in particular in the neocortex.
<figure><img src="image_12.png" /></figure>

Figure B.2: The hippocampal replay step allows engrams reactivation. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). We modeled hippocampal reactivation by sending a spatially uniform stimulus on H. All H neurons receive this input, but only those involved in the pattern fire for a long time, because of their non zero learning weights. Activated H neurons lead to firing in C neurons, which then activate D neurons. (b) Intra and (c) inter-field learning weights at the end of the hippocampal replay phase (solid lines) compared to learning weights at the end of the encoding phase (dotted lines). Weights consolidate thanks to pattern reactivation, in particular in the neocortex.
<figure><img src="image_13.png" /></figure>

Figure B.3: The retrieval cue step allows engrams reactivation. This step consists in stimulating only the $A^C$ or $B^C$ location of the neocortex (here $B^C$). Retrieval is exhibited by subsequent activation of $A^C$. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). A signal is sent on only $B^C$ in the neocortical field. The weights between $A^C$ and $B^C$ are still small, C is still dependent on H to recover the whole pattern. Thus, $B^C$ neurons first activate $B^D$ then $B^H$ neurons, which will activate $A^H$ neurons thanks to the strong enough $B^H-A^H$ learning weights. Finally, the $A^H$ neurons activate $A^C$ neurons via the strong $A^C-A^H$ weights, leading to the whole pattern recovery. (b) Intra and (c) inter-field learning weights at the end of the retrieval cue process (solid lines) compared to learning weights at the end of the encoding phase (dotted lines). Weights are consolidated thanks to pattern reactivation, in particular in the neocortex.
<figure><img src="image_13.png" /></figure>

Figure B.3: The retrieval cue step allows engrams reactivation. This step consists in stimulating only the $A^C$ or $B^C$ location of the neocortex (here $B^C$). Retrieval is exhibited by subsequent activation of $A^C$. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). A signal is sent on only $B^C$ in the neocortical field. The weights between $A^C$ and $B^C$ are still small, C is still dependent on H to recover the whole pattern. Thus, $B^C$ neurons first activate $B^D$ then $B^H$ neurons, which will activate $A^H$ neurons thanks to the strong enough $B^H-A^H$ learning weights. Finally, the $A^H$ neurons activate $A^C$ neurons via the strong $A^C-A^H$ weights, leading to the whole pattern recovery. (b) Intra and (c) inter-field learning weights at the end of the retrieval cue process (solid lines) compared to learning weights at the end of the encoding phase (dotted lines). Weights are consolidated thanks to pattern reactivation, in particular in the neocortex.
<figure><img src="image_14.png" /></figure>

Figure B.4: Neocortical engram reactivation depends on hippocampal fields during the retrieval cue steps. The retrieval cue stimulation with hippocampal “lesion” is obtained by cancelling the learning weights amplitudes in the hippocampal fields, between them and the neocortical field ($\gamma^{\alpha\beta} = 0$ for all $\alpha,\beta$ except $\gamma^{CC}$). (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). Only the $B^\alpha$ neurons, directly stimulated by the partial retrieval cue, fire. Memory cannot be retrieved without a functional hippocampus. (b) The intra and (c) inter-field learning weights between the $A^\alpha$ and $B^\alpha$ cannot be consolidated in absence of firing of the whole pattern.
<figure><img src="image_14.png" /></figure>

Figure B.4: Neocortical engram reactivation depends on hippocampal fields during the retrieval cue steps. The retrieval cue stimulation with hippocampal “lesion” is obtained by cancelling the learning weights amplitudes in the hippocampal fields, between them and the neocortical field ($\gamma^{\alpha\beta} = 0$ for all $\alpha,\beta$ except $\gamma^{CC}$). (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H). Only the $B^\alpha$ neurons, directly stimulated by the partial retrieval cue, fire. Memory cannot be retrieved without a functional hippocampus. (b) The intra and (c) inter-field learning weights between the $A^\alpha$ and $B^\alpha$ cannot be consolidated in absence of firing of the whole pattern.
<figure><img src="image_15.png" /><figcaption>Figure B.5: Neurogenesis in the dentate gyrus prevents the reactivation of hippocampal engrams. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H), during the hippocampal replay phase. Neurons in the neighbourhood of the pattern in the dentate gyrus field are modeled as highly excitable newborn neurons (lower threshold, see the methods subsection 3). The thin lines in the D field show the continuous firing of those new neurons. As a result, the thresholds of the neurons in the vicinity adapt and stay high. Therefore, the neurons of the pattern stay silent in response to the replay pattern. (b) During neocortical retrieval cues, C neurons fire since there are no direct connections from D to C. However nothing fires in the hippocampal fields. The retrieval cue activates the whole pattern in C independently of the hippocampal fields. (c) Intra and (d) inter-field learning weights after 325 cycles with neurogenesis (solid lines). Learning weights are fully consolidated in the neocortex and have disappeared in hippocampal fields.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure B.5: Neurogenesis in the dentate gyrus prevents the reactivation of hippocampal engrams. (a) Firing rates in the three fields, neocortex (C), dentate gyrus (D) and CA regions (H), during the hippocampal replay phase. Neurons in the neighbourhood of the pattern in the dentate gyrus field are modeled as highly excitable newborn neurons (lower threshold, see the methods subsection 3). The thin lines in the D field show the continuous firing of those new neurons. As a result, the thresholds of the neurons in the vicinity adapt and stay high. Therefore, the neurons of the pattern stay silent in response to the replay pattern. (b) During neocortical retrieval cues, C neurons fire since there are no direct connections from D to C. However nothing fires in the hippocampal fields. The retrieval cue activates the whole pattern in C independently of the hippocampal fields. (c) Intra and (d) inter-field learning weights after 325 cycles with neurogenesis (solid lines). Learning weights are fully consolidated in the neocortex and have disappeared in hippocampal fields.</figcaption></figure>