Uncertainty-Dependent Extinction of Fear Memory in an Amygdala-mPFC Neural Circuit Model

Uncertainty of fear conditioning is crucial for the acquisition and extinction of fear memory. Fear memory acquired through partial pairings of a conditioned stimulus (CS) and an unconditioned stimulus (US) is more resistant to extinction than that acquired through full pairings; this effect is known as the partial reinforcement extinction effect (PREE). Although the PREE has been explained by psychological theories, the neural mechanisms underlying the PREE remain largely unclear. Here, we developed a neural circuit model based on three distinct types of neurons (fear, persistent and extinction neurons) in the amygdala and medial prefrontal cortex (mPFC). In the model, the fear, persistent and extinction neurons encode predictions of net severity, of unconditioned stimulus (US) intensity, and of net safety, respectively. Our simulation successfully reproduces the PREE. We revealed that unpredictability of the US during extinction was represented by the combined responses of the three types of neurons, which are critical for the PREE. In addition, we extended the model to include amygdala subregions and the mPFC to address a recent finding that the ventral mPFC (vmPFC) is required for consolidating extinction memory but not for memory retrieval. Furthermore, model simulations led us to propose a novel procedure to enhance extinction learning through re-conditioning with a stronger US; strengthened fear memory up-regulates the extinction neuron, which, in turn, further inhibits the fear neuron during re-extinction. Thus, our models increased the understanding of the functional roles of the amygdala and vmPFC in the processing of uncertainty in fear conditioning and extinction.


Fear memory as a statistical inference
We aimed to reveal the type of statistical processing that animals perform during fear conditioning with partial reinforcement and subsequent extinction. The fact that extinction learning largely depends on the uncertainty of the US during fear conditioning (Fig 2I-K) suggests that the uncertainty must be encoded in the brain. Thus, animals may process the statistical properties of sequential US observations. As statistical processing performed by the animals, we adopted a statistical model based on sequential updating of Bayesian logistic regression (see Statistical inference model below). Because continuous and discontinuous presentation of the US during full and partial reinforcement paradigms leads to fear memories that differ in their resistance to extinction, the effect of US continuity was incorporated into the model; the probability that the US would occur was inferred based on the previous observation of the US (equation (S1)). Then, our simulation demonstrated that, like the basic model, the sequentially predicted US probability reproduced, (Fig 2) the characteristics of the extinction of fear memory acquired through both the full and partial reinforcement schedules (S3C  Fig and S4F Fig). Taken together, these commonalities between the two models suggest that the neural circuit model that consisted of fear, persistent and extinction neurons effectively processed the statistical property of the occurrence of the US through sequential updating of Bayesian logistic regression.

Statistical inference model
To quantify the degree of surprise from the perspective of inferring whether the US would occur, we developed a statistical model based on logistic regression with sequential Bayesian updating [1]. We assumed that the animals inferred the probability that the US would occur based on the previous observation of the US as and sequentially updated w t+1 =[w t+1,0 , w t+1,1 ] to predict the probability based on a new instance of the US, where US t denotes a binary variable, i.e., US=0 and =1 represent the presence and absence of the US, respectively; w t,0 and w t,1 are internal variables in animals, representing the effects of inferring the probability that the US would occur independent of and dependent on the previous US, respectively; is the logistic function σ(x)=1/(1+e −x ); π t represents the probability of the US occurrence that the animals infer based on the previous US, US t-1 . We also assumed that the degree of surprise (S) that animals feel in response to the US or no-US is quantified by the amount of information [2]: (S2) We considered w t+1 to be updated by sequential Bayesian updating as where P(w t |US 1:t-1 ) and P(w t |US 1:t ) represent the prior and posterior distributions, respectively. The animals were assumed to think that instances of the US were simply generated by a Bernoulli process In addition, the animals were assumed to believe that w t remained almost constant with a small degree of noise as w t =w t−1 +ε t , where ε t represents independent and identically distributed Gaussian noise with a mean of zero and a low variance, s. Then, where I represents the unit matrix.
The Bayesian belief update was implemented as an extended Kalman filter, in which the prior and posterior distributions are approximated by Gaussian distributions. P(w t-1 |US 1:t-1 ) was represented by where N represents the function of the Gaussian distribution parameterized by Σ t and μ t , which are the variance-covariance matrix and mean vector of w t , respectively. Then, equation (S3) can be transformed into P(w t |US 1:t ) ∝ P(US t | w t ,US 1:t−1 )N (w t | µ t−1 , s + Σ t−1 ) .
Note that the prior and posterior distributions in equation (S7) do not have conjugate relationship.
Thus, the posterior distribution, P(w t |US 1:t ), was approximated by a Gaussian distribution using Laplace approximation as P(w t |US 1: where μ t and Σ t are updated as where φ indicates (1, US t−1 ) T and (S11) To calculate μ t requires numerical optimization because E(w t ) is a non-linear function of w t .