Synaptic competition in the lateral amygdala and the stimulus specificity of conditioned fear: a biophysical modeling study

Abstract

Competitive synaptic interactions between principal neurons (PNs) with differing intrinsic excitability were recently shown to determine which dorsal lateral amygdala (LAd) neurons are recruited into a fear memory trace. Here, we explored the contribution of these competitive interactions in determining the stimulus specificity of conditioned fear associations. To this end, we used a realistic biophysical computational model of LAd that included multi-compartment conductance-based models of 800 PNs and 200 interneurons. To reproduce the continuum of spike frequency adaptation displayed by PNs, the model included three subtypes of PNs with high, intermediate, and low spike frequency adaptation. In addition, the model network integrated spatially differentiated patterns of excitatory and inhibitory connections within LA, dopaminergic and noradrenergic inputs, extrinsic thalamic and cortical tone afferents to simulate conditioned stimuli as well as shock inputs for the unconditioned stimulus. Last, glutamatergic synapses in the model could undergo activity-dependent plasticity. Our results suggest that plasticity at both excitatory (PN–PN) and di-synaptic inhibitory (PN–ITN and, particularly, ITN–PN) connections are major determinants of the synaptic competition governing the assignment of PNs to the memory trace. The model also revealed that training-induced potentiation of PN–PN synapses promotes, whereas that of ITN–PN synapses opposes, stimulus generalization. Indeed, suppressing plasticity of PN–PN synapses increased, whereas preventing plasticity of interneuronal synapses decreased the CS specificity of PN recruitment. Overall, our results indicate that the plasticity configuration imprinted in the network by synaptic competition ensures memory specificity. Given that anxiety disorders are characterized by tendency to generalize learned fear to safe stimuli or situations, understanding how plasticity of intrinsic LAd synapses regulates the specificity of learned fear is an important challenge for future experimental studies.

Appendices

Appendix

Here, we list additional information related to methods, including the mathematical equations, implementation of the effects of neuromodulators, and the iterative procedures. All model runs were performed using parallel NEURON (Carnevale and Hines 2006) running on a Beowulf supercluster with a time step of 50 μs. Simulation output was analyzed using MATLAB.

Mathematical equations for voltage-dependent ionic currents

The equation for each compartment (soma or dendrite) followed the Hodgkin–Huxley formulations (Byrne and Roberts 2004) in Eq. (1),

$$C_{\text{m}} {\text{d}}V_{\text{s}} /{\text{d}}t = - g_{\text{L}} (V_{\text{s}} - E_{\text{L}} ) - g_{\text{c}} (V_{\text{s}} - V_{\text{d}} ) - \mathop \sum \nolimits I_{\text{cur,s}}^{\text{int}} - \mathop \sum \nolimits I_{{{\text{cur,}}\;{\text{s}}}}^{\text{syn}} + I_{\text{inj}}$$

(1)

where $V_{\text{s}} /V_{\text{d}}$ are the somatic/dendritic membrane potential (mV), $I_{\text{cur,s}}^{\text{int}}$ and $I_{\text{cur,s}}^{\text{syn}}$ are the intrinsic and synaptic currents in the soma, $I_{\text{inj}}$ is the electrode current applied to the soma, $C_{\text{m}}$ is the membrane capacitance, $g_{\text{L}}$ is the conductance of leak channel, $g_{\text{c}} = 1/{\text{Ra}}$ is the coupling conductance between the soma and the dendrite (similar term added for other dendrites connected to the soma), and E _L is the leak reversal potential. Eq. (1) represents a current balance, with the sum of all currents being equal to the injected current. The term on the left represents the capacitance current. The intrinsic current $I_{\text{cur,s}}^{\text{syn}}$, was modeled as $I_{\text{cur,s}}^{\text{int}} = g_{\text{cur}} m^{p} h^{q} (V_{\text{s}} - E_{\text{cur}} )$, where $g_{\text{cur}}$ is its maximal conductance, m its activation variable (with exponent p), h its inactivation variable (with exponent q), and $E_{\text{cur}}$ its reversal potential (a similar equation is used for the synaptic current $I_{\text{cur,s}}^{\text{syn}}$ but without m and h). The kinetic equation for each of the gating variables x (m or h) takes the form

$$\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{{x_{\infty } \;\left( {V,\left[ {{\text{Ca}}^{2 + } } \right]_{i} } \right) - x}}{{\tau_{x} \;\left( {V,\left[ {{\text{Ca}}^{2 + } } \right]_{i} } \right)}}$$

(2)

where $x_{\infty }$ is the steady state gating voltage- and/or Ca²⁺-dependent gating variable and $\tau_{x}$ is the voltage- and/or Ca²⁺ -dependent time constant. The equation for the dendrite follows the same format with ‘s’ and ‘d’ switching positions in Eq. (1). Details related to the model, including types of channels and parameter values are provided in Tables 2 and 3.

Table 2 Gating variables for ion channels used in the single cell models

Table 3 Maximal conductance densities of ion channels

Mathematical equations for synaptic currents

Excitatory transmission was mediated by AMPA/NMDA receptors, and inhibitory transmission by GABA_A receptors. The corresponding synaptic currents were modeled by dual exponential functions (Durstewitz et al. 2000), as shown in Eqs. (3)–(5),

$$I_{\text{AMPA}} = w\left( t \right)\;{ \times }\;G_{\text{AMPA}} \;{ \times }\;(V - E_{\text{AMPA}} )$$

(3)

$$I_{\text{NMDA}} \; = \;w\;{ \times }\;G_{\text{NMDA}} \;{ \times }\;(V - E_{\text{NMDA}} )$$

(4)

$$I_{\text{GABA}} \; = \;w\left( t \right)\;{ \times }\;G_{\text{GABA}} \;{ \times }\;(V - E_{\text{GABA}} )$$

(5)

where V is the membrane potential (mV) of the compartment (dendrite or soma) where the synapse is located and w is the adjustable synaptic weight for the synapse (w was variable for AMPA and GABA synapses, but fixed for NMDA synapses), and G _X is the conductance of the particular synapse (see Sect. “Calcium dynamics and Hebbian learning” for expressions for G _X ). The synaptic reversal potentials were E _AMPA = E _NMDA = 0 mV and E _GABAA = −75 mV (Durstewitz et al. 2000).

Calcium dynamics and Hebbian learning

Intracellular calcium concentration, $\left[ {{\text{Ca}}^{2 + } } \right]_{\text{pool}}$, was regulated by a simple first-order differential equation shown in Eq. (6) (Warman et al. 1994),

$$\frac{{d\left[ {{\text{Ca}}^{2 + } } \right]_{\text{pool}} }}{{{\text{d}}t}} = - f\frac{{I_{\text{Ca}}^{X} }}{zFV} + \frac{{\left[ {{\text{Ca}}^{2 + } } \right]_{\text{rest}} - \left[ {{\text{Ca}}^{2 + } } \right]_{\text{pool}} }}{{\tau_{\text{Ca}} }}$$

(6)

where $I_{\text{Ca}}^{X}$ is the relevant current (NMDA, AMPA, or GABA) contributing to the pool (refer to Eqs. 8–10); f if the fraction of the Ca²⁺ component of the relevant current (f = 0.024); volume V = (4/3 × π × r_pool³) with r_pool = 0.9086 mm, z = 2 is the valence of the Ca²⁺ ion; F is the Faraday constant; and is τ _Ca the time constant associated with Ca²⁺ removal. The resting Ca²⁺ concentration was $\left[ {{\text{Ca}}^{2 + } } \right]_{\text{rest}}$ = 50 nmol/l (Durstewitz et al. 2000).

The biophysical Hebbian rule was implemented by adjusting the synaptic weight w(t) in synaptic conductances (Eqs. 3, 5) using Eq. (7),

$$\Delta w_{j} = \eta \left( {\left[ {{\text{Ca}}^{2 + } } \right]_{j} } \right)\Delta t\left( {\lambda_{1} \Omega \left( {\left[ {{\text{Ca}}^{2 + } } \right]_{j} } \right) - \lambda_{2} \omega_{j} } \right)$$

(7)

where η is the Ca²⁺-dependent learning rate and Ω is a Ca²⁺-dependent function with two thresholds (θ _d and θ _p; θ _d ≤ θ _p) (for details, see Li et al. 2009); λ ₁ and λ ₂ represent scaling and decay factors, respectively; the local calcium level at synapse j is denoted by $\left[ {{\text{Ca}}^{2 + } } \right]_{j}$ and Δt is the simulation time step. With this learning rule, the synaptic weight decreases when θ _d < $\left[ {{\text{Ca}}^{2 + } } \right]_{j}$ < θ _p, and increases when $\left[ {Ca^{2 + } } \right]_{j}$ > θ _p, with modulation by the decay term λ ₂ w _j.

Concentration of calcium pools

The concentration of the calcium pool at synapse j followed the dynamics in Eq. (6), with f _j  = 0.024 (Warman et al. 1994), τ _j  = 50 ms (Shouval et al. 2002b), V is the volume of a spine head with a diameter of 2 μm (Kitajima and Hara 1997). All the synaptic weights were constrained by upper (W _max) and lower (W _min) limits (Li et al. 2009). Maximum (f _max) and minimum (f _min) folds were specified for each modifiable synapse so that W _max = f _max × w(0) and W _min = f _min × w(0).

Excitatory synapses onto principal cells

For tone-PN, and PN–PN connections, the calcium influx $I_{\text{Ca}}^{N}$ which determines learning was estimated as in Li et al. (2009), using Eq. (8),$I_{\text{Ca}}^{N} = P_{0} G_{\text{NMDA}} (V - E_{\text{Ca}} ),$ where

$$\begin{aligned} G_{\text{NMDA}} = g_{\text{NMDA,max}} \;{ \times }\;{\text{STP}}_{\text{NMDA}} { \times }s\left( V \right)\;{ \times }\;r_{\text{NMDA}} \hfill \\ r_{\text{NMDA}} = \alpha T{ \hbox{max} }_{\text{NMDA}} { \times }{\text{ON}}_{\text{NMDA}} { \times }\left( {1 - r_{\text{NMDA}} } \right) - \beta_{\text{NMDA}} { \times }r_{\text{NMDA}} \hfill \\ \end{aligned}$$

(8)

where P ₀ = 0.015, the reversal potential of calcium E _Ca = 120 mV; the maximal conductance for NMDA current $g_{\text{NMDA,max}}$ = 0.5 nS; STP_NMDA is the short-term plasticity factor (see Sect. “Short-term presynaptic plasticity”); the voltage-dependent variable s(V) which implements the Mg²⁺ block was defined as: s(V) = [1 + 0.33 exp(−0.06 V)]⁻¹ (Zador et al. 1990); $r_{\text{NMDA}}$ is the fraction of bound receptors; $\alpha T{ \hbox{max} }_{\text{NMDA}}$ = 0.2659/ms and $\beta_{\text{NMDA}}$ = 0.0008/ms are specific synaptic current parameters; and ON_NMDA = 1 if the NMDA receptor is open, else 0.

Excitatory synapses onto interneurons

For tone-interneuron, and principal cell–interneuron connections, the calcium influx (used for learning) at the excitatory synapses on interneurons occurs through both NMDA and AMPA receptors Eqs. (8) and (9) (details in Li et al. 2009) with P ₀ = 0.001 in Eq. (9).

$I_{\text{Ca}}^{A} = P_{0} wG_{\text{AMPA}} (V - E_{\text{Ca}} )/w(0)$, where

$$\begin{aligned}G_{\text{AMPA}} &= g_{\text{AMPA,max}} \;{ \times }\;{\text{STP}}_{\text{AMPA}} \;{ \times }\;r_{\text{AMPA}} \\ r_{\text{AMPA}} &= \alpha T_{\text{AMPA}} \;{ \times }\;{\text{ON}}_{\text{AMPA}} \;{ \times }\;\left( {1 - r_{\text{AMPA}} } \right) - \beta_{\text{AMPA}} \;{ \times }\;r_{\text{AMPA}} \end{aligned}$$

(9)

where the parameters are as defined in Eq. (8), with P ₀ = 0.001; the maximal conductance for AMPA current $g_{\text{AMPA,max}}$ = 1 nS; $r_{\text{AMPA}}$ is the fraction of bound receptors; $\alpha T{ \hbox{max} }_{\text{AMPA}}$ = 3.8142/ms and $\beta_{\text{AMPA}}$ = 0.1429/ms, and w(0) is the initial weight of the synapse. The Ca²⁺ current through the AMPA/NMDA receptors was separated from the total AMPA/NMDA current in this manner and used for implementation of the learning rule (Kitajima and Hara 1997; Shouval et al. 2002a; Li et al. 2009).

Inhibitory synapses onto principal cells

Several different mechanisms have been reported for potentiation at GABAergic synapses in other brain regions (e.g., Gaiarsa et al. 2002). A rise in postsynaptic intracellular Ca²⁺ concentration seems to be required in most mechanisms to trigger long-term plasticity. In the neonatal rat hippocampus, potentiation could be induced by Ca²⁺ influx through the voltage-dependent Ca²⁺ channels (VDCCs), whereas in the cortex and cerebellum, this process requires Ca²⁺ release from postsynaptic internal stores that is dependent on stimulation of GABA receptors (Gaiarsa et al. 2002). Thus, both presynaptic activity (GABA receptor stimulation or interneuron firing) and postsynaptic activity (activation of VDCCs by membrane depolarization) contribute to the potentiation of GABA synapses. The process from GABA receptor stimulation to internal Ca²⁺ release involves activating a cascade of complex intracellular reactions (Komatsu 1996). Such a complex process can be simplified by assuming that the Ca²⁺ release is proportional to the stimulation frequency or GABA_A conductance (Li et al. 2009). Hence, we modeled this simplified process by considering Ca²⁺ release from the internal stores into a separate Ca²⁺ pool, using an equation similar to that for the AMPA/NMDA case cited above, as shown in Eq. (10), $I_{\text{Ca}}^{G} = P_{0} G_{\text{GABA}} (V - E_{\text{Ca}} )$, where

$$\begin{aligned} G_{\text{GABA}} = g_{\text{GABAmax}} \;{ \times }\;{\text{STP}}_{\text{GABA}} \;{ \times }\;r_{\text{GABA}} \hfill \\ r_{\text{GABA}} = \alpha T{ \hbox{max} }_{\text{GABA}} \;{ \times }\;{\text{ON}}_{\text{GABA}} \;{ \times }\;\left( {1 - r_{\text{GABA}} } \right) - \beta_{\text{GABA}} \;{ \times }\;r_{\text{GABA}} \hfill \\ \end{aligned}$$

(10)

with the parameters again as defined in Eq. (8), with P ₀ = 0.01; the maximal conductance for GABA current $g_{\text{GABA,max}}$ = 0.6 nS; $r_{\text{GABA}}$ is the fraction of bound receptors; $\alpha T{ \hbox{max} }_{\text{GABA}}$ = 7.2609/ms and $\beta_{\text{GABA}}$ = 0.2667/ms.

The current $I_{\text{Ca}}^{G}$ models the dependence of Ca²⁺ release on GABA_A stimulation frequency but not Ca²⁺ influx through the GABA_A channel. $I_{\text{Ca}}^{G}$, together with portion of postsynaptic voltage-dependent calcium current (I _Ca), contributed towards plasticity. The total calcium influx into the pool for learning was $I_{\text{Ca}}^{G} + 0.01I_{\text{Ca}}$ for such synapses (Li et al. 2009).

Intrinsic connectivity in LAd

By comparing the responses of LAd cells to local applications of glutamate at various positions with respect to recorded neurons, Samson and Paré (2006) inferred general principles of connectivity among principal cells, as well as between local-circuit and principal neurons. In particular, Samson and Paré (2006) determined that excitatory connections between principal cells prevalently run ventrally and medially with significant rostrocaudal divergence. In contrast, inhibitory connections prevalently run mediolaterally in the horizontal plane and have no preferential directionality in the coronal plane. Samson and Paré (2006) also recognized that principal LAd neurons located along the external capsule (in the “shell” region of LA) form different connections than those found more medially (in the “core” region of LA; shell thickness of 100 µm). In the shell region, inhibitory neurons only affect nearby principal neurons, whereas excitatory connections between principal cells are spatially less limited. While not providing precise connectivity data, this information could be used to infer critical estimates about directionality and ratio of excitation to inhibition. The directionality information from the Samson and Paré (2006) study, described in the two paragraphs that follow, were implemented in the model, using a third of the connectivity numbers. Such an implementation for one model run showed that a model principal cell had, on average, 21.4 mono- and 40.6 di-synaptic excitatory inputs, and 20.4 mono-synaptic inhibitory inputs.

Coronal plane

Within a 100-µm coronal slice, principal shell neurons excite shell cells located 300–400 µm more ventrally with 10 % probability (mono-synaptic connectivity). Core to shell connections occur with a much lower probability (2 %). In addition, principal shell neurons are inhibited by more dorsally located interneurons (23 % connectivity for cells within 300 μm). In the core region, excitatory connections between principal cells have a greater extent in the lateromedial direction (50–800 μm, 2–6 %, connectivity) than in the mediolateral direction (50–200 μm, 5 % connectivity), whereas inhibitory connections have similar strengths in all directions (interneurons formed inhibitory inputs with 10–24 % of principal cells at a distance of 50–600 μm).

Horizontal plane

Within a 100-µm horizontal slice, connections were set in the following manner. Connection probability increased with distance for lateromedial connections, and the opposite for mediolateral connections. As to inhibitory connections, they prevalently run in the mediolateral direction with 8–20 % connectivity in the range 50–600 μm and 5–20 % connectivity in the lateromedial direction within a distance of 50–600 μm. Principal cells project to all interneurons within a spherical radius of 100 μm. Figures related to these connectivity configurations can be found in Kim et al. (2013a; Fig. 2).

Short-term presynaptic plasticity

Short-term plasticity was implemented as follows (Varela et al. 1997; Hummos et al. 2014): For facilitation, the factor F was calculated using Eq. (11).

$$\tau_{F} { \times }\frac{{{\text{d}}F}}{{{\text{d}}t}} = 1 - F;F\left( 0 \right) = 1, \;{\text{and it is constrained to be}} \ge 1$$

(11)

After each stimulus, F was multiplied by a constant f (≥1) representing the amount of facilitation per presynaptic action potential, and updated as $F \to F{ \times }f$. Between stimuli, F recovered exponentially back toward 1. A similar scheme was used to calculate the factor D for depression,

$$\tau_{Di} { \times }\frac{{{\text{d}}Di}}{{{\text{d}}t}} = 1 - D_{i} ;D_{i} \left( 0 \right) = 1,\;{\text{and it is constrained to be}} \le 1$$

(12)

where i varied from 1 to the number of depression factors, permitting use of different time constants. After each stimulus, D _i was multiplied by a constant d _i (≤1) representing the amount of depression per presynaptic action potential, and updated as $D_{i} \to D_{i} { \times }d_{i}$. Between stimuli, D _i recovered exponentially back toward 1. We modeled depression using two factors d ₁ and d ₂ with d ₁ being fast and d ₂ being slow subtypes. The parameters for the short-term plasticity models, the initial weights and other learning parameters for the synapses are listed in Table 4.

Table 4 Model synaptic strengths and learning parameters

Modeling neuromodulator effects

Blockade of DA and NE has been shown to impair the acquisition of fear memory in LA. These have been modeled by adjusting both intrinsic and synaptic parameters based on experimental reports (see Kim et al. 2013a for details) as shown in Table 5.

Table 5 Variations in maximal conductances to model neuromodulator effects