fix jupyter notebook documentation rendering

doc/tutorials/sequence_learning/sequence_learning.ipynb

@@ -61,7 +61,7 @@
     "\n",
     "$$\n",
     "\\tau_{\\text{m},i} \\frac{d V_{\\text{m},i}(t)}{dt} = -V_{\\text{m},i}(t) + R_{\\text{m},i} I_i(t)\n",
-    "\\label{eq:membrane_potential} \\quad (1)\n",
+    "\\label{eq:membrane_potential} \\quad\\quad (1)\n",
     "$$\n",
     "\n",
     "with membrane resistance $R_{\\text{m},i} = \\tau_{\\text{m},i} C_{\\text{m},i}$, membrane time constant $\\tau_{\\text{m},i}$, and total synaptic input current $I_i(t)$.\n",
@@ -77,14 +77,14 @@
     "\n",
     "$$\n",
     "I_{\\text{ED},i} = \\sum_{j\\in E} (\\alpha_{i,j} \\ast s_j)(t - d_{ij})\n",
-    "\\label{eq:currentED} \\quad (2)\n",
+    "\\label{eq:currentED} \\quad\\quad (2)\n",
     "$$\n",
     "\n",
     "with \n",
     "\n",
     "$$\n",
     "\\alpha_{i,j}(t) = J_{i,j} \\frac{e}{\\tau_\\text{ED}} t e^{-t / \\tau_\\text{ED}} \\Theta(t)\n",
-    "\\label{eq:alpha} \\quad (3)\n",
+    "\\label{eq:alpha} \\quad\\quad (3)\n",
     "$$\n",
     "\n",
     "and\n",
@@ -94,7 +94,7 @@
     "1 & \\text{if $t \\geq 0$} \\\\\n",
     "0 & \\text{else}\n",
     "\\end{cases}\n",
-    "\\quad (4)\n",
+    "\\quad\\quad (4)\n",
     "$$\n",
     "\n",
     "All external, inhibitory and excitatory input currents $I_{\\text{EX},i}(t), I_{\\text{EI},i}(t), I_{\\text{IE},i}(t)$ evolve according to\n",
@@ -105,7 +105,7 @@
     "\\tau_\\text{EI} \\frac{I_{\\text{EI},i}}{dt} &= -I_{\\text{EX},i}(t) + \\sum_{j\\in I} J_{i,j} s_j (t - d_{i,j})\\\\\n",
     "\\tau_\\text{IE} \\frac{I_{\\text{IE},i}}{dt} &= -I_{\\text{EX},i}(t) + \\sum_{j\\in E} J_{i,j} s_j (t - d_{i,j})\\\\\n",
     "\\end{align*}\n",
-    "\\label{eq:tausyn} \\quad (5)\n",
+    "\\label{eq:tausyn} \\quad\\quad (5)\n",
     "$$\n",
     "\n",
     "The dendritic current includes an additional nonlinearity describing the generation of dAPs: if the dendritic current $I_\\text{ED}$ exceeds a threshold $\\theta_\\text{dAP}$, it is instantly set to a the dAP plateau current $I_\\text{dAP}$, and clamped to this value for a period of duration $\\tau_\\text{dAP}$. This plateau current leads\n",
@@ -132,17 +132,17 @@
     "&- \\lambda_- \\sum_{\\{t_j^\\ast\\}} \\delta(t - t_j^\\ast)\\\\\n",
     "&+\\lambda_\\text{h} \\sum_{\\{t_i^\\ast\\}'} (z^\\ast - z_i(t)) \\delta(t - t_i^\\ast) I(t_i^\\ast, \\Delta t_\\text{min}, \\Delta t_\\text{max})\n",
     "\\end{align}\n",
-    "\\label{eq:permanences} \\quad (6)\n",
+    "\\label{eq:permanences} \\quad\\quad (6)\n",
     "$$\n",
     "\n",
-    "The first term on the right-hand side of $\\eqref{eq:permanences}$ corresponds to the spike-timing-dependent synaptic potentiation triggered by the postsynaptic spikes. The indicator function $I(t^\\ast_i, \\Delta t_\\text{min}, \\Delta t_\\text{max})$ ensures that the potentiation (and the homeostasis; see below) is restricted to time lags $t_i^\\ast - t_j^+ + d_\\text{EE}$ in the interval $(\\Delta t_\\text{min}, \\Delta t_\\text{max})$ to avoid a growth of synapses between synchronously active neurons belonging to the same subpopulation, and between neurons encoding for the first elements in different sequences:\n",
+    "The first term on the right-hand side of (6) corresponds to the spike-timing-dependent synaptic potentiation triggered by the postsynaptic spikes. The indicator function $I(t^\\ast_i, \\Delta t_\\text{min}, \\Delta t_\\text{max})$ ensures that the potentiation (and the homeostasis; see below) is restricted to time lags $t_i^\\ast - t_j^+ + d_\\text{EE}$ in the interval $(\\Delta t_\\text{min}, \\Delta t_\\text{max})$ to avoid a growth of synapses between synchronously active neurons belonging to the same subpopulation, and between neurons encoding for the first elements in different sequences:\n",
     "\n",
     "$$\n",
     "I(t_i^\\ast, \\Delta t_\\text{min}, \\Delta t_\\text{max}) = \\begin{cases} \n",
     "1 & \\text{if $\\Delta t_\\text{min} < t_i^\\ast - t_j^+ + d_\\text{EE} < \\Delta t_\\text{max}$} \\\\\n",
     "0 & \\text{else}\n",
     "\\end{cases}\n",
-    "\\label{eq:indicator} \\quad (6)\n",
+    "\\label{eq:indicator} \\quad\\quad (7)\n",
     "$$\n",
     "\n",
     "Note that the potentiation update times lag the somatic postsynaptic spike times by the delay $d_\\text{EE}$, which is here interpreted as a purely dendritic delay (Morrison et al., 2007).\n",
@@ -151,18 +151,18 @@
     "\n",
     "$$\n",
     "\\tau_+ \\frac{dx_j}{dt} = -x_j(t) + \\sum_{t_j^\\ast} \\delta(t - t_j^\\ast)\n",
-    "\\label{eq:pretrace} \\quad (7)\n",
+    "\\label{eq:pretrace} \\quad\\quad (8)\n",
     "$$\n",
     "\n",
     "The trace $x_j(t)$ is incremented by 1 at each spike time $t^∗_j$, followed by an exponential decay with time constant $\\tau_+$. The potentiation increment $\\Delta P_{i,j}$ at time $t^∗_i$ therefore depends on the temporal distance between the postsynaptic spike time $t^∗_i$ and all presynaptic spike times $t^\\ast_j \\leq t^\\ast_i$ (STDP with all-to-all spike pairing [Morrison et al. 2008]). \n",
     "\n",
-    "The second term on the right-hand side of $\\eqref{eq:permanences}$ represents synaptic depression, and is triggered by each presynaptic spike at times $t^\\ast_j \\in \\{t^\\ast_j\\}$. The depression decrement is treated as a constant equal to 1, independently of the postsynaptic spike history. The depression magnitude is parameterized by the dimensionless depression rate $\\lambda_-$.\n",
+    "The second term on the right-hand side of (6) represents synaptic depression, and is triggered by each presynaptic spike at times $t^\\ast_j \\in \\{t^\\ast_j\\}$. The depression decrement is treated as a constant equal to 1, independently of the postsynaptic spike history. The depression magnitude is parameterized by the dimensionless depression rate $\\lambda_-$.\n",
     "\n",
-    "The third term on the right-hand side of $\\eqref{eq:permanences}$ corresponds to a homeostatic control triggered by postsynaptic spikes at times $t^\\ast_i \\in \\{t^\\ast_i\\}'$. Its overall impact is parameterized by the dimensionless homeostasis rate $\\lambda_\\text{h}$. The homeostatic control enhances or reduces the synapse growth depending on the dAP trace $z_i(t)$ of neuron $i$, the low-pass filtered dAP activity updated according to\n",
+    "The third term on the right-hand side of (6) corresponds to a homeostatic control triggered by postsynaptic spikes at times $t^\\ast_i \\in \\{t^\\ast_i\\}'$. Its overall impact is parameterized by the dimensionless homeostasis rate $\\lambda_\\text{h}$. The homeostatic control enhances or reduces the synapse growth depending on the dAP trace $z_i(t)$ of neuron $i$, the low-pass filtered dAP activity updated according to\n",
     "\n",
     "$$\n",
     "\\tau_\\text{h}\\frac{dz_i}{dt} = -z_i(t) + \\sum_k \\delta(t - t^k_{\\text{dAP},i})\n",
-    "\\label{eq:lpfdAP} \\quad (8)\n",
+    "\\label{eq:lpfdAP} \\quad\\quad (9)\n",
     "$$\n",
     "\n",
     "Synapse growth is boosted if the dAP activity $z_i(t)$ is below a target dAP activity $z^\\ast$. Conversely, high dAP activity exceeding $z^\\ast$ reduces the synapse growth.\n",