equations.ipynb

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    "# Membrane potential\n",
    "\n",
    "We use leaky integrate-and-fire neurons with conductance-based synapses (see Section 13.6.2 of Gerstner, Kistler, Naud, and Paninski (http://neuronaldynamics.epfl.ch/index.html)).\n",
    "\n",
    "The $i$-th neuron's membrane voltage $V^i(t)$ is given by:\n",
    "\n",
    "$\\tau_m\\cfrac{dV^i}{dt} = -(V^i - E_L) - g^i_E(t)(V^i - E_E) - g^i_I(t)(V^i - E_I) + I^i_{ext}(t)$\n",
    "\n",
    "where $\\tau_m$ is the membrane time constant, $E_L$ is the leak conductance, $g^i_E$ and $E_E$ the excitatory conductance and reversal potential, respectively, and $g^i_I$ and $E_I$ the inhibitory conductance and reversal potential, with conductances defined relative to the leak conductance.\n",
    "\n",
    "When the voltage of neuron $i$ crosses the threshold $V_{th}^i$ it spikes and is reset to $V_r^i$ for $\\tau_{rp}^i$ seconds."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Associated model params\n",
    "\n",
    "$\\tau_m^{PC}$ (T_M_PC): PC (pyramidal/place cell) membrane time constant\n",
    "\n",
    "$E_L^{PC}$ (E_L_PC): PC membrane leak potential\n",
    "\n",
    "$V_{th}^{PC}$ (V_TH_PC): PC spike threshold\n",
    "\n",
    "$V_r^{PC}$ (V_R_PC): PC reset potential\n",
    "\n",
    "$\\tau_{r}^{PC}$ (T_R_PC): PC refractory period\n",
    "\n",
    "\n",
    "$\\tau_m^{INH}$ (T_M_INH): INH membrane time constant\n",
    "\n",
    "$E_L^{INH}$ (E_L_INH): INH membrane leak potential\n",
    "\n",
    "$V_{th}^{INH}$ (V_TH_INH): INH spike threshold\n",
    "\n",
    "$V_r^{INH}$ (V_R_INH): INH reset potential\n",
    "\n",
    "$\\tau_{r}^{INH}$ (T_R_INH): INH refractory period\n",
    "\n",
    "\n",
    "\n",
    "$E_E$ (E_E): EXC synaptic reversal potential\n",
    "\n",
    "$E_I$ (E_I): INH synaptic reversal potential"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Synaptic conductances\n",
    "\n",
    "The synaptic conductances for the $i$-th cell are given by:\n",
    "\n",
    "$g^i_{SYN}(t) = \\sum\\limits_j w_{ij} \\sum\\limits_{k(j)} h_{SYN}(t - t^{k(j)})$\n",
    "\n",
    "where $g^i_{SYN} \\in \\{g^i_E, g^i_I\\}$, $j$ indexes upstream neurons, $w_{ij}$ is the synaptic strength from neuron $j$ onto neuron $i$, $k(j)$ is the $k$-th spike time of upstream neuron $j$, and $h_{SYN}(t)$ is the synaptic activation function for synapse type $SYN$ (in this case a one-sided exponential with time constant $\\tau_{SYN}$). These conductances are relative to the leak conductance and have no units.\n",
    "\n",
    "In short, the synaptic conductance of neuron $i$ caused by neuron $j$'s spike train is a convolution of neuron $j$'s spike train (where each spike is a delta function) with synaptic activation function $h(t)$ (e.g., a 1-sided decaying exponential), scaled by the synaptic weight $w_{ij}$.\n",
    "\n",
    "__Note__: the synaptic weights $w_{ij}$ given in the manuscript are relative to the leak conductance. In the code, however, they have been scaled by an additional factor of $\\Delta t/\\tau_m$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Associated model params\n",
    "\n",
    "$\\tau_E$ (T_E): EXC synaptic decay time constant\n",
    "\n",
    "$\\tau_I$ (T_I): INH synaptic decay time constant"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Place-tuned inputs\n",
    "\n",
    "$\\bar{w}^{PC, PL}$ (W_PC_PL): mean EXC PL->PC synaptic weight\n",
    "\n",
    "$s^{PC, PL}$ (S_PC_PL): std EXC PL->PC synaptic weight"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# GATE --> PC connectivity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All gate --> PC connectivity is purely excitatory."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Associated model params"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\bar{w}^{PC, G}$ (W_PC_G): mean EXC synaptic weight to PC from its associated gate input\n",
    "\n",
    "$s^{PC, G}$ (S_PC_G): std EXC synaptic weight to PC from its associated gate input"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# GATE --> PC inputs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PCs receive poisson spiking inputs from corresponding upstream gate neurons. These occur at one rate during the sensory/traj epoch and at another rate during the replay epoch."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Associated model params\n",
    "\n",
    "$r^G$ (R_G): gating unit spike rate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# GATE --> PC activity-dependent plasticity\n",
    "\n",
    "Hyun (2013, 2015) showed that a 10 Hz spike train in a CA3 PC lasting 1-2 s, elicited either by direct current injection or activation of upstream mossy fibers, was sufficient to almost double the size of EC->CA3 EPSPs.\n",
    "\n",
    "We model this activity-dependent synaptic weight change in the $i$-th neuron by scaling all G-->PC weights $w_{ij}$ by an activity dependent scaling factor $\\sigma^i = \\sigma(r^i)$, where $r^i$ location-evoked firing rate in PC $i$. Thus, $w_{ij} \\rightarrow \\sigma^iw_{ij}$ following the agent's trajectory through its environment, where $j$ idxes gating neurons."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The location-evoked firing rate $r^i$ is a squared exponential function of the distance from PC $i$'s place field center and the nearest point on the agent's trajectory.\n",
    "\n",
    "$r^i = r(d^i) = r_{max}^{PL} \\exp\\left(-\\cfrac{1}{2}\\left(\\cfrac{d^i}{\\lambda_{PL}}\\right)^2\\right)$\n",
    "\n",
    "The scaling factor $\\sigma^i$ on $w^{ij}$ $\\forall j$ is \n",
    "\n",
    "$\\sigma^i = \\sigma(r^i) = 1 + \\cfrac{\\sigma_{max} - 1}{1 + \\exp[-\\beta_\\sigma(r^i - r_\\sigma)]}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Associated model params\n",
    "\n",
    "$r_{max}^{PL}$ (R_MAX_PL): firing rate at place-field center\n",
    "\n",
    "$\\lambda^{PL}$ (L_PL): place-field length scale\n",
    "\n",
    "$\\sigma_{max}$ (SGM_MAX): max amplification factor for G-->PC EXC weight change\n",
    "\n",
    "$\\beta_{\\sigma}$ (B_SGM): sigmoid steepness\n",
    "\n",
    "$r_\\sigma$ (R_SGM): sigmoid offset (threshold location)\n",
    "\n",
    "$\\bar{w}^{PC,G}$ (W_PC_G): mean G-->PC cxn weight\n",
    "\n",
    "$s^{PC,G}$ (S_PC_G): std G-->PC cxn weight"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PC->PC connectivity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We model recurrent connectivity among CA3 PCs by assuming that cells with similar place-tuning have a higher probability of connecting. Specifically, the probability of a candidate targ PC receiving a connection from a src PC at a distance $d$ away is given by:\n",
    "\n",
    "$w^{PC,PC}(d) = \\bar{w}^{PC, PC}\\exp\\left[-\\cfrac{1}{2}\\left(\\cfrac{d}{\\lambda^{PC,PC}}\\right)^2\\right]$\n",
    "\n",
    "All $w^{PC,PC} < w_{min}^{PC,PC}$ set to 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Associated model params\n",
    "\n",
    "$N^{PC}$ (N_PC): num PCs in ntwk\n",
    "\n",
    "$\\bar{w}^{PC, PC}$ (W_PC_PC): max PC-->PC synaptic weight\n",
    "\n",
    "$\\lambda^{PC, PC}$ (L_PC_PC): PC-->PC connectivity length scale\n",
    "\n",
    "$w_{min}^{PC,PC}$ (W_MIN_PC_PC): min nonzero PC-->PC syn weight"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reduced model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Voltage dynamics:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$v^i(t) = \\sum_j w_{ij} s_j(t - 1) + i_{ext}(t) + i_g \\sigma^i$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Intersecting trajectory"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Connectivity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\n",
    "w(\\Delta x, \\Delta y, \\Delta \\theta) = \n",
    "\\exp\\left(-\\cfrac{\\Delta x^2 + \\Delta y^2}{2\\lambda_{xy}^2}\\right)\n",
    "\\exp\\left(-\\cfrac{\\Delta \\theta^2}{2\\lambda_\\theta}\\right)\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "LTP-IE profile:\n",
    "\n",
    "$\\sigma(x, y, \\theta) = 1 + \\cfrac{1}{1 + \\exp[-\\beta(z-z_0)]}$\n",
    "\n",
    "where\n",
    "\n",
    "$z(x, y, \\theta) = \\exp\\left[-.5\\left(\\cfrac{\\Delta x^2}{\\sigma_x^2} + \\cfrac{\\Delta y^2}{\\sigma_y^2} + \\cfrac{\\Delta \\theta}{\\sigma_{\\theta}^2}\\right)\\right]$"
   ]
  }
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