Neural dynamics of cross-modal and cross-temporal associations

Abstract

We have studied a neurodynamic model of cross-modal and cross-temporal associations. We show that a network of integrate-and-fire neurons can generate spiking activity with realistic dynamics during the delay period of a paired associates task. In particular, the activity of the model resembles reported data from single-cell recordings in the prefrontal cortex.

Appendices

Appendix 1

We used the mathematical formulation of the IF neurons and synaptic currents described in Brunel and Wang (2001). Here we provide a brief summary of their framework. The dynamics of the sub-threshold membrane potential V of a neuron are given by the equation:

$$ C_{{\text{m}}} \frac{{{\text{d}}V(t)}} {{{\text{d}}t}}=- g_{{\text{m}}} (V(t) - V_{{\text{L}}} ) - I_{{{\text{syn}}}} (t), $$

where C_m is the membrane capacitance, taken to be 0.5 nF for excitatory neurons and 0.2 nF for inhibitory neurons; g_m is the membrane leak conductance, taken to be 25 nS for excitatory neurons and 20 nS for inhibitory neurons; V_L is the resting potential of −70 mV and I_syn is the synaptic current. The firing threshold is taken to be V_thr=−50 mV and the reset potential V_reset=−55 mV.

The synaptic current is given by a sum of glutamatergic, AMPA (I_AMPA,rec) and NMDA (I_NMDA,rec) mediated, recurrent excitatory currents, one AMPA (I_AMPA,ext) mediated external excitatory current, and one inhibitory GABAergic current (I_GABA):

$$ I_{{{\text{syn}}}} (t)=I_{{{\text{AMPA}},{\text{ext}}}} (t)+I_{{{\text{AMPA}},{\text{rec}}}} (t)+I_{{{\text{NMDA}},{\text{rec}}}} (t)+I_{{{\text{GABA}}}} (t). $$

The currents are defined by:

$$ I_{{{\text{AMPA}},{\text{ext}}}} (t)=g_{{{\text{AMPA}},{\text{ext}}}} (V(t) - V_{E} ){\sum\limits_{j=1}^{N_{{{\text{ext}}}} } {s^{{{\text{AMPA}},{\text{ext}}}}_{j} (t)} } $$

$$ I_{{{\text{AMPA}},{\text{rec}}}} (t)=g_{{{\text{AMPA}},{\text{rec}}}} (V(t) - V_{E} ){\sum\limits_{j=1}^{N_{E} } {w_{j} s^{{{\text{AMPA}},{\text{rec}}}}_{j} (t)} } $$

$$ I_{{{\text{NMDA}},{\text{rec}}}} (t)=\frac{{g_{{{\text{NMDA}}}} (V(t) - V_{E} )}} {{1+[Mg^{{++ }} ]\exp ( - 0.062V(t))/3.57}} \times {\sum\limits_{j=1}^{N_{E} } {w_{j} s^{{{\text{NMDA}}}}_{j} (t)} } $$

$$ I_{{{\text{GABA}}}} (t)=g_{{{\text{GABA}}}} (V(t) - V_{I} ){\sum\limits_{j=1}^{N_{I} } {s^{{{\text{GABA}}}}_{j} (t)} } $$

where V _E =0 mV, V _I =−70 mV, w _j are the synaptic weights, s _j the fractions of open channels for the different receptors, and g the synaptic conductances for the different channels. The NMDA synaptic current depends on the membrane potential and is controlled by the extracellular concentration of magnesium ([Mg²⁺]=1 mmol L⁻¹). The values for the synaptic conductances for excitatory neurons are g_AMPA,ext=2.08 nS, g_{AMPA, rec}=0.104 nS, g_NMDA=0.327 nS, and g_GABA=1.25 nS and for inhibitory neurons g_AMPA,ext=1.62 nS, g_AMPA,rec=0.081 nS, g_NMDA=0.258 nS, and g_GABA=0.973 nS. These values are the same as in Brunel and Wang (2001). In their work the conductances were calculated so that in an unstructured network the excitatory neurons have a spontaneous spiking rate of 3 Hz and the inhibitory neurons a spontaneous rate of 9 Hz. The fractions of open channels are described by:

$$ \frac{{{\text{d}}s^{{{\text{AMPA}},{\text{ext}}}}_{j} (t)}} {{{\text{d}}t}}=- \frac{{s^{{{\text{AMPA}},{\text{ext}}}}_{j} (t)}} {{\tau _{{{\text{AMPA}}}} }}+{\sum\limits_k {\delta {\left( {t - t^{k}_{j} } \right)}} } $$

$$ \frac{{{\text{d}}s^{{{\text{AMPA}},{\text{rec}}}}_{j} (t)}} {{{\text{d}}t}}=- \frac{{s^{{{\text{AMPA}},{\text{rec}}}}_{j} (t)}} {{\tau _{{{\text{AMPA}}}} }}+{\sum\limits_k {\delta {\left( {t - t^{k}_{j} } \right)}} } $$

$$\frac{{{\text{d}}s^{{{\text{NMDA}}}}_{j} (t)}} {{{\text{d}}t}}=- \frac{{s^{{{\text{NMDA}}}}_{j} (t)}} {{\tau _{{{\text{NMDA}},{\text{decay}}}} }}+\alpha x_{j} (t){\left( {1 - s^{{{\text{NMDA}}}}_{j} (t)} \right)} $$

$$\frac{{{\text{d}}x_{j} (t)}} {{{\text{d}}t}}=- \frac{{x_{j} (t)}} {{\tau _{{{\text{NMDA}},{\text{rise}}}} }}+{\sum\limits_k {\delta {\left( {t - t^{k}_{j} } \right)}} } $$

$$ \frac{{{\text{d}}s^{{{\text{GABA}}}}_{j} (t)}} {{{\text{d}}t}}=- \frac{{s^{{{\text{GABA}}}}_{j} (t)}} {{\tau _{{{\text{GABA}}}} }}+{\sum\limits_k {\delta {\left( {t - t^{k}_{j} } \right)}} }, $$

where τ_NMDA,decay, =100 ms, is the decay time for NMDA synapses, τ_AMPA, =2 ms, the decay time for AMPA synapses, and τ_GABA, =10 ms, the decay time for GABA synapses; τ_NMDA,rise, =2 ms, is the rise time for NMDA synapses (the rise times for AMPA and GABA are neglected because they are smaller than 1 ms) and α=0.5 ms⁻¹. The sums over k represent a sum over spikes formulated as δ-Peaks (δ(t)) emitted by presynaptic neuron j at time t ^k _j . The equations were integrated numerically using the second order Runge–Kutta method with step size 0.05 ms.

Appendix 2

The mean-field approximation used in this work was derived from Brunel and Wang 2001, assuming that the network of integrate-and-fire neurons is in a stationary state. In this formulation the potential of a neuron is calculated as:

$$ \tau _{x} \frac{{{\text{d}}V(t)}} {{{\text{d}}t}}=- V(t)+\mu _{x}+\sigma _{x} {\sqrt {\tau _{x} } }\eta (t) $$

where V(t) is the membrane potential, x labels the populations, τ _x is the effective membrane time constant, μ _x is the mean value the membrane potential would have in the absence of spiking and fluctuations, σ _x measures the magnitude of the fluctuations, and η is a Gaussian process with absolute exponentially decaying correlation function with time constant τ_AMPA. The quantities μ _x and σ ² _x are given by:

$$ \mu _{x}=\frac{{{\left( {T_{{{\text{ext}}}} \nu _{{{\text{ext}}}}+T_{{{\text{AMPA}}}} n_{x}+\rho _{1} N_{x} } \right)}V_{E}+\rho _{2} N_{x} {\left\langle V \right\rangle }+T_{I} w_{{I,x}} \nu _{I} V_{I}+V_{L} }} {{S_{x} }} $$

$$ \sigma ^{2}_{x}=\frac{{{\left( {g^{2}_{{{\text{AMPA}},{\text{ext}}}} \nu _{{{\text{ext}}}}+g^{2}_{{{\text{AMPA}},{\text{rec}}}} \nu _{x} } \right)}{\left( {{\left\langle V \right\rangle } - V_{E} } \right)}^{2} \tau ^{2}_{{{\text{AMPA}}}} \tau _{x} }}{{g^{2}_{m} \tau ^{2}_{m} }}.$$

where w_I,x are the weights from the inhibitory neurons to the pool x, ν _ext is the external impinging spiking rate (including spontaneously activity and eventually external stimuli or attentional bias), ν _I is the population-averaged spiking rate of the inhibitory pool, τ _m =C _m /g _m with the values for the excitatory or inhibitory neurons depending on the pool considered, and the other quantities are given by:

$$ S_{x}=1+T_{{{\text{ext}}}} \nu _{{{\text{ext}}}}+T_{{A{\text{MPA}}}} n_{x}+(\rho _{1}+\rho _{2} )N_{x}+T_{I} w_{{I,x}} \nu _{I} $$

$$ \tau _{x}=\frac{{C_{{\text{m}}} }} {{g_{m} S_{x} }} $$

$$ n_{x}={\sum\limits_{j=1}^p {f_{j} w_{{j,x}} \nu _{j} } } $$

$$N_{x}={\sum\limits_{j=1}^p {f_{j} w_{{j,x}} \psi (\nu _{j} )} }$$

$$ \psi (\nu )=\frac{{\nu \tau _{{{\text{NMDA}}}} }} {{1+\nu \tau _{{{\text{NMDA}}}} }}{\left( {1+\frac{1} {{1+\nu \tau _{{{\text{NMDA}}}} }}{\sum\limits_{n=1}^\infty {\frac{{{\left( { - \alpha \tau _{{{\text{NMDA}},{\text{rise}}}} } \right)}^{n} T_{n} (\nu )}} {{(n+1)!}}} }} \right)} $$

$$ T_{n} (\nu )={\sum\limits_{k=0}^n {( - 1)^{k} {\left( {\begin{array}{*{20}c} {n} \\ {k} \\ \end{array} } \right)}} }\frac{{\tau _{{{\text{NMDA}},{\text{rise}}}} (1+\nu \tau _{{{\text{NMDA}}}} )}} {{\tau _{{{\text{NMDA}},{\text{rise}}}} (1+\nu \tau _{{{\text{NMDA}}}})+k\tau _{{{\text{NMDA}},{\text{decay}}}}}} $$

$$ \tau _{{{\text{NMDA}}}}=\alpha \tau _{{{\text{NMDA}},{\text{rise}}}} \tau _{{{\text{NMDA}},{\text{decay}}}} $$

$$ T_{{{\text{ext}}}}=\frac{{g_{{{\text{AMPA}},{\text{ext}}}} \tau _{{{\text{AMPA}}}} }} {{g_{m} }} $$

$$ T_{{{\text{AMPA}}}}=\frac{{g_{{{\text{AMPA}},{\text{rec}}}} N_{E} \tau _{{{\text{AMPA}}}} }} {{g_{m} }} $$

$$ \rho _{1}=\frac{{g_{{{\text{NMDA}}}} N_{E} }} {{g_{m} J}} $$

$$ \rho _{2}=\beta \frac{{g_{{{\text{NMDA}}}} N_{E} {\left( {{\left\langle {V_{x} } \right\rangle } - V_{E} } \right)}(J - 1)}} {{g_{m} J^{2} }} $$

$$ J=1+\gamma \exp {\left( { - \beta {\left\langle {V_{x} } \right\rangle }} \right)} $$

$$ T_{I}=\frac{{g_{{{\text{GABA}}}} N_{I} \tau _{{{\text{GABA}}}} }} {{g_{m} }} $$

$$ {\left\langle {V_{x} } \right\rangle }=\mu _{x} - {\left( {V_{{{\text{thr}}}} - V_{{{\text{reset}}}} } \right)}\nu _{x} \tau _{x} , $$

(1)

where p is the number of excitatory pools, f _x is the fraction of neurons in the excitatory x pool, w_j,x the weight of the connections from pool x to pool j, ν _x is the population-averaged spiking rate of the x excitatory pool, γ=1/3.57, β=0.062 and the average membrane potential 〈V _x 〉 has a value between −55 mV and −50 mV.

The spiking rate of a pool as a function of the defined quantities is then given by:

$$ \nu _{x}=\phi {\left( {\mu _{x} ,\sigma _{x} } \right)}, $$

(2)

where

$$ \phi {\left( {\mu _{{x,}} \sigma _{x} } \right)}={\left( {\tau _{{{\text{rp}}}}+\tau _{x} {\int_{\beta (\mu _{x} ,\alpha _{x} )}^{\alpha (\mu _{x} ,\alpha _{x} )} {{\text{d}}u{\sqrt \pi }\exp (u^{2} )[1+{\text{erf}}(u)]} }} \right)}^{{ - 1}} $$

$$ \alpha {\left( {\mu _{x} ,\sigma _{x} } \right)}=\frac{{{\left( {V_{{{\text{thr}}}} - \mu _{x} } \right)}}} {{\sigma _{x} }}{\left( {1+0.5\frac{{\tau _{{{\text{AMPA}}}} }} {{\tau _{x} }}} \right)}+1.03{\sqrt {\frac{{\tau _{{{\text{AMPA}}}} }} {{\tau _{x} }}} } - 0.5\frac{{\tau _{{{\text{AMPA}}}} }} {{\tau _{x} }} $$

$$ \beta {\left( {\mu _{x} ,\sigma _{x} } \right)}=\frac{{{\left( {V_{{{\text{reset}}}} - \mu _{x} } \right)}}} {{\sigma _{x} }} $$

where erf(u) is the error function and τ_rp the refractory period, which is regarded as 2 ms for excitatory neurons and 1 ms for inhibitory neurons. To solve the equations defined by Eq. (2) for all values of x we integrate Eq. (1) numerically and the differential equation below, describing a fake dynamics of the system, which has fixed point solutions corresponding to Eqs. (2):

$$ \tau _{x} \frac{{{\text{d}}\nu _{x} }} {{{\text{d}}t}}=- \nu _{x}+\phi {\left( {\mu _{x} ,\sigma _{x} } \right)}. $$

(3)

For all simulation periods studied, the mean-field equations, Eqs. (1) and (3) were integrated using the Euler method with step size 0.1 and 4,000 iterations.