Effects of heterogeneity in synaptic conductance between weakly coupled identical neurons

Abstract

A significant degree of heterogeneity in synaptic conductance is present in neuron to neuron connections. We study the dynamics of weakly coupled pairs of neurons with heterogeneities in synaptic conductance using Wang–Buzsaki and Hodgkin–Huxley model neurons which have Types I and II excitability, respectively. This type of heterogeneity breaks a symmetry in the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant when compared to the identical case. For weakly coupled neurons coupled with identical values of synaptic conductance a phase locked solution exists for all values of the synaptic rate constant, α. In particular, in-phase and anti-phase solutions are guaranteed to exist for all α. Heterogeneity in synaptic conductance results in regions where no phase locked solution exists and the general loss of the ubiquitous in-phase and anti-phase solutions of the identically coupled case. We explain these results through examination of interaction functions using the weak coupling approximation and an in-depth analysis of the underlying multiple cusp bifurcation structure of the systems of coupled neurons.

Appendices

Appendix A: Hodgkin–Huxley model

The equations for the Hodgkin–Huxley Model are:

$$\begin{array}{rll} C_m\frac{dV}{dt}&=&-g_Kn^4(V-E_K)-g_{\rm Na}m^3h(V-E_{\rm Na})\\ &&-\,g_L(V-E_L)+I_{\rm stim} \end{array}$$

(7)

$$ \frac{dn}{dt}=\alpha_n(1-n)-\beta_nn $$

(8)

$$ \frac{dm}{dt}=\alpha_m(1-m)-\beta_mm $$

(9)

$$ \frac{dh}{dt}=\alpha_h(1-h)-\beta_hh $$

(10)

$$ \alpha_n=0.01\frac{V+55}{1-\exp{\frac{-(V+55)}{10}}} $$

(11)

$$ \beta_n=0.125\exp{\frac{-(V+65)}{80}} $$

(12)

$$ \alpha_m=0.1\frac{V+40}{1-\exp{\frac{-(V+40)}{10}}} $$

(13)

$$ \beta_m=4.0\exp{\frac{-(V+65)}{18}} $$

(14)

$$ \alpha_h=0.07\exp{\frac{-(V+65)}{20}} $$

(15)

$$ \beta_h=\frac{1}{1-\exp{\frac{-(V+35)}{10}}} $$

(16)

With $g_{\rm Na}=120\:\frac{\rm mS}{{\rm cm}^2}$, $g_K=36\:\frac{\rm mS}{{\rm cm}^2}$, $g_L=0.3\:\frac{\rm mS}{{\rm cm}^2}$, $E_{\rm Na}=50\:{\rm mV}$, $E_K=-77\:{\rm mV}$, $V_L=-54.6\:{\rm mV}$, $C_m=1\:\frac{\mu \rm F}{{\rm cm}^2}$, $I_{\rm stim}=10\:\frac{\mu \rm A}{{\rm cm}^2}$.

Appendix B: Wang–Buzsaki model

The equations for the Wang–Buzsaki Model are:

$$ \begin{array}{rll} C_m\frac{dV}{dt}&=&-g_Kn^4(V-E_K)-g_{\rm Na}m_{\infty}^3h(V-E_{\rm Na})\\ &&-\,g_L(V-E_L)+I_{\rm stim} \end{array} $$

(17)

$$ \frac{dn}{dt}=\phi(\alpha_n(1-n)-\beta_nn) $$

(18)

$$ \frac{dh}{dt}=\phi(\alpha_h(1-h)-\beta_hh) $$

(19)

$$ m_{\infty}=\frac{\alpha_m}{\alpha_m+\beta_m} $$

(20)

$$ \alpha_m=\frac{-0.1(V+35)}{\exp{-0.1(V+35)}-1} $$

(21)

$$ \beta_m=4\exp{-(V+60)/18} $$

(22)

$$ \alpha_h=0.07\exp{-(V+58)/20} $$

(23)

$$ \beta_h=\frac{1}{\exp{-0.1(V+28)}+1} $$

(24)

$$ \alpha_n=\frac{-0.1(V+34)}{\exp{-0.1(V+34)}-1} $$

(25)

$$ \beta_n=0.125\exp{-(V+44)/80} $$

(26)

With $g_{\rm Na}=35\:\frac{\rm mS}{{\rm cm}^2}$, $g_K=9\:\frac{\rm mS}{{\rm cm}^2}$, $g_L=0.1\:\frac{\rm mS}{{\rm cm}^2}$, $E_{\rm Na}=55\:{\rm mV}$, $E_K=-90\:{\rm mV}$, $V_L=-65\:{\rm mV}$, $C_m=1\:\frac{\mu \rm F}{{\rm cm}^2}$, $I_{\rm stim}=10\:\frac{\mu \rm A}{{\rm cm}^2}$, φ = 5.