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Fix bug with
integrate_odes() for numeric solver (nest#1147)
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tests/nest_tests/resources/aeif_cond_alpha_alt_neuron.nestml
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| """ | ||
| aeif_cond_alpha - Conductance based exponential integrate-and-fire neuron model | ||
| ############################################################################### | ||
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| Description | ||
| +++++++++++ | ||
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| aeif_psc_alpha is the adaptive exponential integrate and fire neuron according to Brette and Gerstner (2005), with post-synaptic conductances in the form of a bi-exponential ("alpha") function. | ||
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| The membrane potential is given by the following differential equation: | ||
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| .. math:: | ||
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| C_m \frac{dV_m}{dt} = | ||
| -g_L(V_m-E_L)+g_L\Delta_T\exp\left(\frac{V_m-V_{th}}{\Delta_T}\right) - | ||
| g_e(t)(V_m-E_e) \\ | ||
| -g_i(t)(V_m-E_i)-w + I_e | ||
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| and | ||
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| .. math:: | ||
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| \tau_w \frac{dw}{dt} = a(V_m-E_L) - w | ||
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| Note that the membrane potential can diverge to positive infinity due to the exponential term. To avoid numerical instabilities, instead of :math:`V_m`, the value :math:`\min(V_m,V_{peak})` is used in the dynamical equations. | ||
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| References | ||
| ++++++++++ | ||
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| .. [1] Brette R and Gerstner W (2005). Adaptive exponential | ||
| integrate-and-fire model as an effective description of neuronal | ||
| activity. Journal of Neurophysiology. 943637-3642 | ||
| DOI: https://doi.org/10.1152/jn.00686.2005 | ||
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| See also | ||
| ++++++++ | ||
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| iaf_psc_alpha, aeif_psc_exp | ||
| """ | ||
| model aeif_cond_alpha_alt_neuron: | ||
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| state: | ||
| V_m mV = E_L # Membrane potential | ||
| w pA = 0 pA # Spike-adaptation current | ||
| refr_t ms = 0 ms # Refractory period timer | ||
| g_exc nS = 0 nS # AHP conductance | ||
| g_exc' nS/ms = 0 nS/ms # AHP conductance | ||
| g_inh nS = 0 nS # AHP conductance | ||
| g_inh' nS/ms = 0 nS/ms # AHP conductance | ||
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| equations: | ||
| inline V_bounded mV = min(V_m, V_peak) # prevent exponential divergence | ||
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| g_exc'' = -2 * g_exc' / tau_syn_exc - g_exc / tau_syn_exc**2 | ||
| g_inh'' = -2 * g_inh' / tau_syn_inh - g_inh / tau_syn_inh**2 | ||
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| # Add inlines to simplify the equation definition of V_m | ||
| inline exp_arg real = (V_bounded - V_th) / Delta_T | ||
| inline I_spike pA = g_L * Delta_T * exp(exp_arg) | ||
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| V_m' = (-g_L * (V_bounded - E_L) + I_spike - g_exc * (V_bounded - E_exc) - g_inh * (V_bounded - E_inh) - w + I_e + I_stim) / C_m | ||
| w' = (a * (V_bounded - E_L) - w) / tau_w | ||
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| refr_t' = -1e3 * ms/s # refractoriness is implemented as an ODE, representing a timer counting back down to zero. XXX: TODO: This should simply read ``refr_t' = -1 / s`` (see https://github.com/nest/nestml/issues/984) | ||
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| parameters: | ||
| # membrane parameters | ||
| C_m pF = 281.0 pF # Membrane Capacitance | ||
| refr_T ms = 2 ms # Duration of refractory period | ||
| V_reset mV = -60.0 mV # Reset Potential | ||
| g_L nS = 30.0 nS # Leak Conductance | ||
| E_L mV = -70.6 mV # Leak reversal Potential (aka resting potential) | ||
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| # spike adaptation parameters | ||
| a nS = 4 nS # Subthreshold adaptation | ||
| b pA = 80.5 pA # Spike-triggered adaptation | ||
| Delta_T mV = 2.0 mV # Slope factor | ||
| tau_w ms = 144.0 ms # Adaptation time constant | ||
| V_th mV = -50.4 mV # Threshold Potential | ||
| V_peak mV = 0 mV # Spike detection threshold | ||
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| # synaptic parameters | ||
| tau_syn_exc ms = 0.2 ms # Synaptic Time Constant Excitatory Synapse | ||
| tau_syn_inh ms = 2.0 ms # Synaptic Time Constant for Inhibitory Synapse | ||
| E_exc mV = 0 mV # Excitatory reversal Potential | ||
| E_inh mV = -85.0 mV # Inhibitory reversal Potential | ||
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| # constant external input current | ||
| I_e pA = 0 pA | ||
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| input: | ||
| exc_spikes <- excitatory spike | ||
| inh_spikes <- inhibitory spike | ||
| I_stim pA <- continuous | ||
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| output: | ||
| spike | ||
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| update: | ||
| if refr_t > 0 ms: | ||
| # neuron is absolute refractory, do not evolve V_m | ||
| integrate_odes(g_exc, g_inh, w, refr_t) | ||
| else: | ||
| # neuron not refractory | ||
| integrate_odes(g_exc, g_inh, V_m, w) | ||
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| onReceive(exc_spikes): | ||
| g_exc' += exc_spikes * (e / tau_syn_exc) * nS * s | ||
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| onReceive(inh_spikes): | ||
| g_inh' += inh_spikes * (e / tau_syn_inh) * nS * s | ||
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| onCondition(refr_t <= 0 ms and V_m >= V_th): | ||
| # threshold crossing | ||
| refr_t = refr_T # start of the refractory period | ||
| V_m = V_reset | ||
| w += b | ||
| emit_spike() |
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