# Model Mathematics

### Component: synaptic_coupling

$sum_g_syn_e_s = g_syn_e_1_2 ⁢ s1 + g_syn_e_1_3 ⁢ s1 + g_syn_e_1_4 ⁢ s1 + g_syn_e_1_5 ⁢ s1 + g_syn_e_2_1 ⁢ s2 + g_syn_e_2_3 ⁢ s2 + g_syn_e_2_4 ⁢ s2 + g_syn_e_2_5 ⁢ s2 + g_syn_e_3_1 ⁢ s3 + g_syn_e_3_2 ⁢ s3 + g_syn_e_3_4 ⁢ s3 + g_syn_e_3_5 ⁢ s3 + g_syn_e_4_1 ⁢ s4 + g_syn_e_4_2 ⁢ s4 + g_syn_e_4_3 ⁢ s4 + g_syn_e_4_5 ⁢ s4 + g_syn_e_5_1 ⁢ s5 + g_syn_e_5_2 ⁢ s5 + g_syn_e_5_3 ⁢ s5 + g_syn_e_5_4 ⁢ s5$

### Component: membrane

$dd time V =- i_NaP + i_Na + i_K + i_L + i_tonic_e + i_syn_e + i_app C$

### Component: fast_sodium_current

$i_Na = g_Na ⁢ m_infinity 3.0 ⁢ 1.0 - n ⁢ V - E_Na$

### Component: fast_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: fast_sodium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: potassium_current

$i_K = g_K ⁢ n 4.0 ⁢ V - E_K$

### Component: potassium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: persistent_sodium_current

$i_NaP = g_NaP ⁢ m_infinity ⁢ h ⁢ V - E_Na$

### Component: persistent_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: persistent_sodium_current_h_gate

$dd time h = h_infinity - h tau_h h_infinity = 1.0 1.0 +ⅇ V - theta_h sigma_h tau_h = tau_h_max cosh⁡ V - theta_h 2.0 ⁢ sigma_h$

### Component: leakage_current

$i_L = g_L ⁢ V - E_L$

### Component: tonic_current

$i_tonic_e = g_tonic_e ⁢ V - E_syn_e$

### Component: synaptic_input

$i_syn_e = sum_g_syn_e_s ⁢ V - E_syn_e s_infinity = 1.0 1.0 +ⅇ V - theta_s sigma_s s = 1.0 - s ⁢ s_infinity -- kr ⁢ s tau_s$

### Component: synaptic_coupling

$sum_g_syn_e_s=g_syn_e_1_2⁢s1+g_syn_e_1_3⁢s1+g_syn_e_1_4⁢s1+g_syn_e_1_5⁢s1$

### Component: membrane

$dd time V =- i_NaP + i_Na + i_K + i_L + i_tonic_e + i_syn_e + i_app C$

### Component: fast_sodium_current

$i_Na = g_Na ⁢ m_infinity 3.0 ⁢ 1.0 - n ⁢ V - E_Na$

### Component: fast_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: fast_sodium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: potassium_current

$i_K = g_K ⁢ n 4.0 ⁢ V - E_K$

### Component: potassium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: persistent_sodium_current

$i_NaP = g_NaP ⁢ m_infinity ⁢ h ⁢ V - E_Na$

### Component: persistent_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: persistent_sodium_current_h_gate

$dd time h = h_infinity - h tau_h h_infinity = 1.0 1.0 +ⅇ V - theta_h sigma_h tau_h = tau_h_max cosh⁡ V - theta_h 2.0 ⁢ sigma_h$

### Component: leakage_current

$i_L = g_L ⁢ V - E_L$

### Component: tonic_current

$i_tonic_e = g_tonic_e ⁢ V - E_syn_e$

### Component: synaptic_input

$i_syn_e = sum_g_syn_e_s ⁢ V - E_syn_e s_infinity = 1.0 1.0 +ⅇ V - theta_s sigma_s s = 1.0 - s ⁢ s_infinity -- kr ⁢ s tau_s$

### Component: synaptic_coupling

$sum_g_syn_e_s=g_syn_e_1_2⁢s1+g_syn_e_1_3⁢s1+g_syn_e_1_4⁢s1+g_syn_e_1_5⁢s1$

### Component: membrane

$dd time V =- i_NaP + i_Na + i_K + i_L + i_tonic_e + i_syn_e + i_app C$

### Component: fast_sodium_current

$i_Na = g_Na ⁢ m_infinity 3.0 ⁢ 1.0 - n ⁢ V - E_Na$

### Component: fast_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: fast_sodium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: potassium_current

$i_K = g_K ⁢ n 4.0 ⁢ V - E_K$

### Component: potassium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: persistent_sodium_current

$i_NaP = g_NaP ⁢ m_infinity ⁢ h ⁢ V - E_Na$

### Component: persistent_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: persistent_sodium_current_h_gate

$dd time h = h_infinity - h tau_h h_infinity = 1.0 1.0 +ⅇ V - theta_h sigma_h tau_h = tau_h_max cosh⁡ V - theta_h 2.0 ⁢ sigma_h$

### Component: leakage_current

$i_L = g_L ⁢ V - E_L$

### Component: tonic_current

$i_tonic_e = g_tonic_e ⁢ V - E_syn_e$

### Component: synaptic_input

$i_syn_e = sum_g_syn_e_s ⁢ V - E_syn_e s_infinity = 1.0 1.0 +ⅇ V - theta_s sigma_s s = 1.0 - s ⁢ s_infinity -- kr ⁢ s tau_s$

### Component: synaptic_coupling

$sum_g_syn_e_s=g_syn_e_1_2⁢s1+g_syn_e_1_3⁢s1+g_syn_e_1_4⁢s1+g_syn_e_1_5⁢s1$

### Component: membrane

$dd time V =- i_NaP + i_Na + i_K + i_L + i_tonic_e + i_syn_e + i_app C$

### Component: fast_sodium_current

$i_Na = g_Na ⁢ m_infinity 3.0 ⁢ 1.0 - n ⁢ V - E_Na$

### Component: fast_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: fast_sodium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: potassium_current

$i_K = g_K ⁢ n 4.0 ⁢ V - E_K$

### Component: potassium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: persistent_sodium_current

$i_NaP = g_NaP ⁢ m_infinity ⁢ h ⁢ V - E_Na$

### Component: persistent_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: persistent_sodium_current_h_gate

$dd time h = h_infinity - h tau_h h_infinity = 1.0 1.0 +ⅇ V - theta_h sigma_h tau_h = tau_h_max cosh⁡ V - theta_h 2.0 ⁢ sigma_h$

### Component: leakage_current

$i_L = g_L ⁢ V - E_L$

### Component: tonic_current

$i_tonic_e = g_tonic_e ⁢ V - E_syn_e$

### Component: synaptic_input

$i_syn_e = sum_g_syn_e_s ⁢ V - E_syn_e s_infinity = 1.0 1.0 +ⅇ V - theta_s sigma_s s = 1.0 - s ⁢ s_infinity -- kr ⁢ s tau_s$

### Component: synaptic_coupling

$sum_g_syn_e_s=g_syn_e_1_2⁢s1+g_syn_e_1_3⁢s1+g_syn_e_1_4⁢s1+g_syn_e_1_5⁢s1$

### Component: membrane

$dd time V =- i_NaP + i_Na + i_K + i_L + i_tonic_e + i_syn_e + i_app C$

### Component: fast_sodium_current

$i_Na = g_Na ⁢ m_infinity 3.0 ⁢ 1.0 - n ⁢ V - E_Na$

### Component: fast_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: fast_sodium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: potassium_current

$i_K = g_K ⁢ n 4.0 ⁢ V - E_K$

### Component: potassium_current_n_gate

$dd time n = n_infinity - n tau_n n_infinity = 1.0 1.0 +ⅇ V - theta_n sigma_n tau_n = tau_n_max cosh⁡ V - theta_n 2.0 ⁢ sigma_n$

### Component: persistent_sodium_current

$i_NaP = g_NaP ⁢ m_infinity ⁢ h ⁢ V - E_Na$

### Component: persistent_sodium_current_m_gate

$m_infinity = 1.0 1.0 +ⅇ V - theta_m sigma_m$

### Component: persistent_sodium_current_h_gate

$dd time h = h_infinity - h tau_h h_infinity = 1.0 1.0 +ⅇ V - theta_h sigma_h tau_h = tau_h_max cosh⁡ V - theta_h 2.0 ⁢ sigma_h$

### Component: leakage_current

$i_L = g_L ⁢ V - E_L$

### Component: tonic_current

$i_tonic_e = g_tonic_e ⁢ V - E_syn_e$

### Component: synaptic_input

$i_syn_e = sum_g_syn_e_s ⁢ V - E_syn_e s_infinity = 1.0 1.0 +ⅇ V - theta_s sigma_s s = 1.0 - s ⁢ s_infinity -- kr ⁢ s tau_s$

### Component: synaptic_coupling

$sum_g_syn_e_s=g_syn_e_1_2⁢s1+g_syn_e_1_3⁢s1+g_syn_e_1_4⁢s1+g_syn_e_1_5⁢s1$
Source
Derived from workspace Butera, Rinzel, Smith II 1999 at changeset 5ffca16220b3.
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