doi:10.1093/brain/awt257
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BRAIN
A JOURNAL OF NEUROLOGY
OCCASIONAL PAPER
Circular inferences in schizophrenia
Renaud Jardri1,2,3 and Sophie Denève1
1 Group for Neural Theory, INSERM U960, Institute of Cognitive Studies, École Normale Supérieure, 75005 Paris, France
2 University Medical Centre of Lille (CHULille), Pediatric Psychiatry Dpt., Fontan Hospital, 59037 Lille, France
3 Lab. de Neurosciences Fonctionnelles and Pathologies, Univ Droit et Santé Lille (UDSL), 59000 Lille, France
Correspondence to: Renaud Jardri,
Group for Neural Theory,
Laboratoire de Neurosciences Cognitives,
INSERM U960, Département d’Études Cognitives,
École Normale Supérieure, 29 rue d’Ulm,
75005 Paris, France
E-mail: [email protected]
A considerable number of recent experimental and computational studies suggest that subtle impairments of excitatory to
inhibitory balance or regulation are involved in many neurological and psychiatric conditions. The current paper aims to
relate, specifically and quantitatively, excitatory to inhibitory imbalance with psychotic symptoms in schizophrenia.
Considering that the brain constructs hierarchical causal models of the external world, we show that the failure to maintain
the excitatory to inhibitory balance results in hallucinations as well as in the formation and subsequent consolidation of
delusional beliefs. Indeed, the consequence of excitatory to inhibitory imbalance in a hierarchical neural network is equated
to a pathological form of causal inference called ‘circular belief propagation’. In circular belief propagation, bottom-up sensory
information and top-down predictions are reverberated, i.e. prior beliefs are misinterpreted as sensory observations and vice
versa. As a result, these predictions are counted multiple times. Circular inference explains the emergence of erroneous percepts,
the patient’s overconfidence when facing probabilistic choices, the learning of ‘unshakable’ causal relationships between unrelated events and a paradoxical immunity to perceptual illusions, which are all known to be associated with schizophrenia.
Keywords: Bayesian; network; belief; predictive coding; hallucinations; delusions
Abbreviations: GABA = gamma-aminobutyric acid; LOR = log-odd ratio; NMDA = N-methyl-D-aspartate
Introduction
Schizophrenia is a devastating mental disorder that afflicts 1% of
the world’s population during their lifetime (McGrath et al., 2008).
The disorder can be defined as a complex array of non-specific
symptoms that can be clustered into at least two to three different
clinical dimensions (Crow, 1981; Liddle, 1987): the ‘positive symptoms’, which include reality distortions such as hallucinations,
delusions and other bizarre beliefs; the ‘negative symptoms’,
comprising psychomotor poverty, blunt affects, social withdrawal
and lack of motivation; and ‘cognitive disorganization’, encompassing dissociation, illogical speech and behaviours.
Although the exact underlying pathophysiological mechanisms
for this disorder remain enigmatic, a substantial amount of physiological (Uhlhaas and Singer, 2010; Mulert et al., 2011) and postmortem (Lewis et al., 2005) evidence converges to suggest an
impairment of gamma-aminobutyric acid (GABA) transmission or
of N-methyl-D-aspartate (NMDA) receptor plasticity in schizophrenia (Stephan et al., 2009). These findings have contributed to a
re-conceptualization of this disorder as a possible disruption in the
neural excitatory to inhibitory balance (O’Donnell, 2011).
Evidence of an excitatory to inhibitory imbalance in schizophrenia includes recent support for an alteration in the inhibitory
GABAergic transmission of cortical microcircuits, comprising
Received March 27, 2013. Revised July 12, 2013. Accepted July 22, 2013. Advance Access publication September 24, 2013
ß The Author (2013). Published by Oxford University Press on behalf of the Guarantors of Brain. All rights reserved.
For Permissions, please email: [email protected]
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dopaminergic, glutamatergic and other GABAergic pathways
(Lisman et al., 2008). Using rodent models, the experimental
blockage of parvalbumin interneurons (Bartos et al., 2002), or
the suppression of their activity using optogenetic methods
(Sohal et al., 2009) was shown to induce significant reductions
in -oscillations, an impairment replicated several times in schizophrenia (for review see Uhlhaas and Singer, 2010). Using induced
pluripotent stem cell technology, another group evidenced general
defects in neuronal connectivity, also compatible with the excitatory to inhibitory imbalance hypothesis (Brennand et al., 2011). By
creating differentiated neurons from patients with schizophrenia,
these authors effectively demonstrated impairments in neuronal
glutamate receptor expression.
This excitatory to inhibitory imbalance was shown to specifically
occur when mesocortical dopaminergic (dopamine) pathways were
activated, resulting in schizophrenia-related phenomena in rodents
(Gruber et al., 2010). This phenomenon of GABAergic interneurons recruitment by dopamine remains inefficient during the
juvenile period (Tseng and O’Donnell, 2007), suggesting that insults of this system would only manifest at latter developmental
stages (Insel, 2010). This mechanism supports the classical emergence period of first episodes of schizophrenia during adolescence
or early adulthood. Interestingly, the reduced GABAergic inhibition
observed in patients with schizophrenia was found to be related to
perceptual deficits (Yoon et al., 2010), such as reduced vulnerability to contrast-contrast illusions (Dakin et al., 2005).
Similarly to reduced GABAergic transmission, the downregulation of NMDA receptors may also affect excitatory to inhibitory
balance. In animal models, global NMDA receptor hypofunction
causes an increase in intrinsic pyramidal cell excitability and a selective disruption of parvalbumin-expressing interneurons (Gandal
et al., 2012). Psychotomimetic models of psychosis, especially
those based on ketamine, also support the excitatory to inhibitory
imbalance hypothesis. Schizophrenia-like symptoms have been
described for ketamine use (Krystal et al., 2005) and in autoimmune anti-NMDA receptor encephalitis (Dalmau et al., 2008).
Specifically, the drug was shown to bind D2R and induce striatal
dopamine release (Smith et al., 1998). Corlett et al. (2009a) suggested that under ketamine, the subject may experience both perceptual aberrations (due to AMPA upregulation) and a reduced
capacity to accommodate and ignore these aberrations (due to
NMDA blockade).
However, the mechanism by which excitatory to inhibitory dysfunctions at the cellular level relates to the abovementioned heterogeneous clinical symptoms of schizophrenia remains unclear.
We consider here that the specific dimensions of schizophrenia
are more reliable constructs than the whole disorder and specifically focus on the potential underlying mechanisms for positive
symptoms, which have been shown to cause severe distress in
>70% of affected individuals (Andreasen and Flaum, 1991). The
question may thus be simplified as to how a failure of the excitatory to inhibitory balance regulation may result in erroneous percepts (i.e. hallucinations) or inflexible beliefs that are not directly
corroborated by the senses (i.e. delusions). Intuitively, less inhibition may result in too much excitation or propagation within cortical circuits, but this hypothesis on its own seems too
reductionistic. Here, we propose moving beyond this relatively
R. Jardri and S. Denève
vague concept by relating impaired inhibition to aberrant belief
formation, considering hierarchical Bayesian inference as a basic
principle of brain function (Friston, 2008).
The Bayesian formalism interprets perception as the inference
on what may have caused the sensory observations, optimally
combining information from sensory receptors with predictions
based on prior knowledge (von Helmholtz, 1866; Knill and
Richards, 1996; Doya et al., 2007). This framework accounts for
erroneous percepts in a non-pathological context, such as perceptual illusions (Geisler and Kersten, 2002; Weiss et al., 2002).
Imbalances between the gain (or confidence) attributed to
bottom-up sensory information compared with top-down prior
knowledge provide a simple account not only for the positive
symptoms of schizophrenia (Friston, 2005a), but also for sensory
integration deficits in autistic spectrum disorders (Pellicano and
Burr, 2012) and functional motor and sensory symptoms in hysteria (Edwards et al., 2012).
Interestingly, the role of inhibitory loops in the brain mechanisms that address ambiguity was recently experimentally strengthened by van Loon et al. (2013). These authors showed that the
modulation of GABAA receptors by the administration of a benzodiazepine to healthy subjects, induced changes in both the excitatory to inhibitory balance, which were measurable through
magnetic resonance spectroscopy of the visual cortex, and the
dynamics of bistable perception.
In the current paper, we show that the maintenance of a precise
excitatory to inhibitory balance is crucial to the implementation of
Bayesian inference in a recurrent biological neural network, and, in
particular, in the proper adjustment of the effective gain of (or
confidence in) feed-forward sensory information versus top-down
priors. A dominance of excitation results in a pathological form of
inference called ‘circular belief propagation’, in which ‘bottom-up’
and ‘top-down’ messages are reverberated and taken into account
multiple times. We show how this concept can parsimoniously
account for hallucinations and delusions while remaining compatible with recent theoretical and empirical findings. Overall, the
presented theory is quantitative and thus testable through behavioural or neurophysiological paradigms.
Materials and methods
Hierarchical inference and excitatory
to inhibitory balance
The brain can be seen as constructing a representation of the world as
a hierarchy of progressively more abstract causes. As a simplified toy
example, let us imagine the processing of evidence for the colour
‘green’ in a visual scene. Green could be caused by the presence of
a leaf. Leaves are themselves predicted by the presence of a tree.
Meanwhile, the presence of a tree could be due to the observer
being located in a forest (Fig. 1A). Hierarchical Bayesian inference
interprets sensory inputs by finding the set of causes that best accounts for sensory observations. This type of inference is a result of
optimally combining two types of information: ‘bottom-up’ (magenta
arrows in Fig. 1) and ‘top-down’ (violet arrows in Fig. 1).
Circular inferences in schizophrenia
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Figure 1 Belief propagation in a hierarchical neural network. (A) Toy example of a hierarchical causal model with six hidden variables
represented as ‘nodes’. Arrows represent causal relationships. Sensory evidence is provided at the bottom and prior evidence at the top of
the hierarchy. (B) Neural network that implements causal inference in this toy example. Mxy represents the message sent by node x to
node y. The overall flow of messages in the network is under the control of inhibitory loops (yellow lines: feed-forward connections; blue
lines: feedback connections; green circles: upward inhibitory loops; black circles: downward inhibitory loops). (C) Belief propagation in the
intact and impaired networks. Left: normal network that performs belief propagation. Because of intact inhibitory loops, ascending
messages that correspond to sensory likelihoods (magenta) and descending messages that correspond to prior expectations (violet) are
propagated only once in each direction, appropriately sharing prior and sensory information among all of the neurons. Right: impaired
network without inhibitory loops. Top-down and bottom-up messages are uncontrollably propagated multiple times in each direction,
which results in the multiple overcounting of the same redundant sensory and prior information. (D) Consequences of circular belief
propagation. The belief obtained through iterating the network dynamics is out of proportion with the available sensory and prior
evidence. This construct results in overconfidence (i.e. probabilities only slightly above 0.5 are perceived as being close to 1, and
probabilities just below 0.5 are perceived as being close to zero).
Bottom-up processing sends information from low-level sensory observations to higher levels. For example, detecting the colour green
increases the probability of a leaf and, in turn, of a tree. Top-down
processing sends prior expectations from high-level causes to their
lower-level sensory consequences. For example, prior knowledge of
being located in a forest increases the expectations of observing
a tree, a leaf and the colour green. In the Bayesian formalism, topdown information corresponds to the prior, and bottom-up information corresponds to the likelihood of sensory inputs. Because this
hierarchy of progressively more abstract representations is loosely
reflected in the hierarchy of sensory-to-associative brain areas,
bottom-up and top-down processing can be interpreted as ‘feedforward’ and ‘feedback’ neural processing (Fig 1A and B; Lee and
Mumford, 2003; Lochmann and Deneve, 2011).
The necessity of combining bottom-up and top-down information
raises a crucial (albeit usually disregarded) issue: that of disentangling
‘new’ and ‘old’ information in the constant flow of neural inputs. The
simultaneous presence of feed-forward and feedback connections creates multiple potential loops in the corresponding neural network.
Without efficient control mechanisms, top-down information, after
descending the hierarchy, could be reverberated back up, resulting
in top-down expectations being misunderstood as ‘real’ sensory observations. The opposite is also true: bottom-up sensory information
could be reverberated back down and misinterpreted as top-down
expectations. As a result, sensory evidence and priors would be propagated many times in the network and accounted for many times
rather than only once. This problem is illustrated in Fig. 1C.
Considering our toy example, walking in the forest can generate expectations for leaves and the colour green. In the presence of loops,
the system could misinterpret the resulting internally generated
expectations as external sensory information, causing the aberrant perception of green leaves, even during winter.
The cortex is a highly recurrent structure in which most long-range
connections are excitatory, and a large proportion of these connections are top-down (Douglas et al., 1995). In these conditions, circular
inference though a reverberation of activity appears to be unavoidable
(Fig. 1C). Fortunately, in spite of the apparent prevalence of excitatory
connections, the balance between excitation and inhibition is tightly
maintained at all levels of neural processing, from single dendrites
and neurons to entire brain areas (Wehr and Zador, 2003; Hensch
and Fagiolini, 2004; Shu et al., 2007; Okun and Lampl, 2008).
Moreover, stimulus-driven or spontaneous temporal modulations of
excitatory and inhibitory drive are temporally correlated (Shu et al.,
2003; Wehr and Zador, 2003). In a recurrent network with large
numbers of feed-forward and feedback excitatory connections, excitatory to inhibitory balance only exists if each excitatory loops is compensated by the presence of an equally strong inhibitory loop (Fig. 1B).
As shown in the next section and illustrated in Fig. 1B and C, such
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R. Jardri and S. Denève
inhibitory loops can predict and cancel out redundant information that
reverberates through both top-down and bottom-up excitatory loops,
avoiding circular inferences. This mechanism ensures that sensory information and prior expectations are accounted for only once, and not
multiple times, in spite of the strong level of recurrence in the
network.
We can now start to see how perturbations of the excitatory to
inhibitory balance result in aberrant belief formation. Indeed, scaling
down these inhibitory loops (or scaling up excitation without an
equivalent increase in inhibition) results in circular inferences and the
generation of erroneous percepts. Moreover, these predictions are
quantifiable (regardless of their specific neural implementation) because lowering the strength of inhibition relative to excitation results
in circular propagation of beliefs in the corresponding normative causal
model, thus predicting specific perceptual and behavioural impairments. In the following section, we present, in more technical terms,
the belief propagation algorithm, its neural interpretation and the proposed role of inhibitory loops.
Belief propagation and its neural
analogy
For the sake of simplicity, we present here a belief propagation algorithm specific to binary causes. ‘Binary’ implies that the causes can
take only two states, for example, a leaf can be either present or
absent. Moreover, we assume that each of the binary variables has
at most one cause in the layer above. We chose binary variables for
illustration purposes and because of their direct interpretability in terms
of decisions, choices and confidence. We assumed only one parent to
keep equations in the main text as simple as possible. However, our
approach readily generalizes to other hierarchical causal models, as
shown in the Supplementary material.
The belief propagation algorithm works by propagating ‘beliefs’,
which can be understood as the current estimate for the probability
for each cause. To facilitate an interpretation in terms of neural processing, we express these beliefs as a log-odd ratio (LOR). As an example, the LOR associated with ‘tree’ is the log of the ratio between
the probability that the tree is present and the probability that the tree
is absent. If the tree is as likely to be present as absent (and thus, the
state of ‘tree’ is completely uncertain), its LORlogð1Þ ¼ 0. On the
other hand, if the tree is three times more likely to be present than
absent, its LOR is logð3Þ 1:1. More generally, LORs are large and
positive if there is a high confidence in state ‘1’ (tree present), they are
large and negative if there is a high confidence in state ‘0’ (tree
absent), and they are close to zero when the state is uncertain. We
call the LORs of cause i at step nBni .
To illustrate the consequences of circular inference, we used a binary
causal graph with four causal layers and six variables (Fig. 1A and B).
Except when otherwise stated, we set the conditional probabilities to
p1ij 0:9 (the probability that j is present given that i is present in 0.9)
and p0ij 0:1 (the probability that j is absent given that i is absent in
0.1) for all of the connected causal pairs. For example, the presence of
a tree causes the presence of a leaf with a probability of 0.9, while the
probability of observing a leaf in the absence of a tree is 0.1. Note that
the qualitative effects reported here are not dependent on the exact
setting of p1ij and p0ij .
Sensory information is provided as messages clamped in the sensory
layer. For example, M5 = 3 implies that there is strong sensory evidence for the colour green (i.e. the probability of the colour green
based on sensory information alone is
e3
1þe3
0:95). M6 = 0 implies that
there is no sensory information available about the presence or absence of a stem (i.e. the probability of the presence of a stem based
on sensory information alone is 0.5). Prior expectations are implemented as a constant message clamped in the top layer. For example,
M5 = 1 implies that the prior probability of being in the forest
(before taking into account sensory information) is
e1
1þe1
0:27. We
assumed constant, fixed priors and likelihoods for the sake of simplicity. Readers should note that the results are similar when ‘noisy’
sensory evidence is provided as random samples from the corresponding likelihood distributions.
Beliefs, Bni , can be computed iteratively by propagating messages
between causal nodes. These messages convey predictions of one
node by another. For example, the message sent from ‘leaf’ to ‘tree’
can be understood as the strength of the evidence for ‘tree’ given the
current belief for ‘leaf’. We will call the message sent from node i to
node j at step nMnij . Initial messages and beliefs are set to zero.
Sensory evidence (i.e. the sensory evidence for green) is provided by
clamping messages in the bottom layer (see toy examples in the
next section). Similarly, prior expectations (i.e. the prior probability
of being in a forest) are implemented by messages in the top
layer. Other messages and beliefs are initialized at zero. Belief propagation implements the following recursive equations until
convergence:
¼ Wij Bni Mnij
Mnþ1
ji
X
Bnþ1
¼
Mnþ1
i
ji
j
The belief propagation algorithm is warranted to converge to the a
posteriori probability distribution for each cause. For example, at the
end of convergence, the belief associated with ‘tree’ is the posterior
probability of a tree given sensory information and prior knowledge. It
is clear from these equations that belief propagation is analogous to
the dynamics of a recurrent neural network, with the activity of single
units interpreted as beliefs that are associated with the variable that
they represent (Fig. 1B). Wij can be understood as representing the
strength of the connection between neuron i and neuron j. Note that
Wij is not a single synaptic weight; instead, it is a sigmoid function
whose shape depends on the conditional probability of cause j, given
cause i. More exactly, it depends on the conditional probabilities that
link node j and node i, p1ij ¼ p xi ¼ 1jxj ¼ 1 and p0ij ¼ pðxi ¼
1jxj ¼ 0Þ, as follows (see Supplementary material for details).
0
1
p1ij eB þ p0ij
A
Wij ðBÞ ¼ log@
1 p1ij eB þ 1 p0ij
is a function of the belief of
Most importantly, the message Mnþ1
ji
the presynaptic node, Bni , corrected by the message previously sent in
the opposite direction. This structure prevents the reverberation of
beliefs between node i and node j by predicting out the redundant
messages. We propose specifically that this correction is implemented
by inhibitory loops, as shown in Fig. 1B.
Circular belief propagation
We parameterize the strength of the inhibitory loops by a scale factor
, between 0 (no inhibition at all) and 1 (normal level of inhibition).
This scaling factor is applied to the message correction, i.e.
Mjinþ1 ¼ Wij Bni Mnij . We also investigate the contributions of different types of inhibitory loops by distinguishing between ‘downward
loops’ for redundant messages descending the hierarchy and ‘upward
Circular inferences in schizophrenia
loops’ for redundant messages climbing the hierarchy. Thus, we distinguish two scale factors for the inhibition of top-down and bottomup messages:
If i is above j:
Mnþ1
¼ Wij Bni d Mnij
ji
If j is above i:
Mnþ1
¼ Wij Bni c Mnij
ji
c and d parameterize the scaling of inhibition for climbing (upward)
and descending (downward) loops, respectively. Thus, downward
loops can be seen as removing from bottom-up ‘sensory’ messages
the redundant ‘top-down’ predictions by higher level areas. The opposite is also true: upward loops can be seen as removing from ‘topdown’ messages the redundant ‘bottom-up’ information from lower
areas that has been already been accounted for.
Learning the precision of the causal
links
As in all hierarchical inference problems solved by the brain, we distinguish between the inference of the unknown states of the world
(binary hidden causes, such as ‘trees’ and ‘leaves’) and the parameters
mediating causal dependencies (e.g. what is the probability of a leaf
given the presence of a tree). Belief propagation computes the probability of the hidden causes on a fast time scale after exposure to a
single stimulus. In contrast, the learning of the causal dependencies
occurs over a slower timescale, during exposure to multiple stimuli. We
will demonstrate how abnormalities of inference lead to severe
abnormalities in learning.
Uncertainty not only is at the heart of decision strategies, but also
controls the learning of internal predictive models (Dayan and Yu,
2003). For example, if a high level of confidence for ‘green’ often
co-occurs with a high level of confidence for ‘stem’, then there is
support for the existence of a higher order common cause for these
two observations, e.g. a ‘leaf’ (Fig. 1A). Here, however, we will not
consider the learning of latent variables or of the structure of the
graph; instead, we will concentrate on the much simpler problem of
learning the strength of causal relationships, or precision with which
one variable predicts another, i.e. the parameters p1ij and p0ij , from
experience.
To illustrate the impact of circular inferences on learning causal relationships from experience, we compared the result of learning based
on the beliefs that are computed by the intact and circular belief
propagation networks (i.e. belief propagation and circular belief propagation, respectively). For illustration purposes, we only learned the
connection between ‘tree’ and ‘leaves’, setting other connections to
their true values. We also used a simplified toy example, without the
variables ‘root’ and ‘stem’. Thus, in this simplified network, ‘forest’
caused the presence of a tree with probability p1true
and p0true
12
12 , ‘tree’
and p0true
caused the presence of a leaf with probability p1true
23
23 , and
and p0true
‘tree’ caused the presence of green with probabilities p1true
35
35 .
We initialized the network connections with parameters p112 ¼ p1true
12 ,
1
1true
0
0true
p012 ¼ p0true
12 , p35 ¼ p35 , and p35 ¼ p35 , whereas the unknown (i.e.
to be learned) connections between ‘tree’ and ‘leaf’ p123 and p023 were
set to a random value that was close to 0.5 (Supplementary material).
For each training example, we sampled the presence or absence of a
tree randomly from a probability of 0.5 and then sampled tree, leaves
and green from their true causal probabilities. ‘Tree’ and ‘green’ were
assumed to be completely observable [i.e. their beliefs were set to
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(0,1) if present and (1,0) if absent]. We relied on belief propagation
or circular belief propagation to compute the beliefs of the hidden
variables (i.e. ‘tree’ and ‘leaves’) and then applied a learning algorithm
called ‘expectation maximization’ (Supplementary material) to update
the estimates for p123 and p023 . Note that the results are not crucially
dependent of the learning algorithm. In the Supplementary material
and Supplementary Fig. 2, we show that a less standard (but more
biologically plausible) Hebbian learning rule leads to the same qualitative effects.
Results
We describe here the main impairments in causal inference and
perceptual decisions as predicted by the circular belief propagation
algorithm. Interpretations of these impairments, both in terms
of the positive symptoms of schizophrenia as well as the other
dimensions of this disorder, are provided in the ‘Discussion’
section.
Normal and circular inference
We first consider the case in which upward and downward loops
are equally impaired, i.e. when c ¼ d < 1. The impairment of
inhibitory loops causes redundant top-down and bottom-up information to be propagated multiple times in the network (Fig. 1C
and D). Quite surprisingly, however, this circular belief propagation does not generate completely ‘false’ percepts, i.e. the signs of
LORs are unaffected (Fig. 1D). For example, if the presence of a
tree is more likely than its absence according to a normal inference, then it is also found to be more likely by the impaired network. This pattern is illustrated with the binary toy example in
Fig. 2A and B. In Fig. 2A, strong sensory evidence is provided
for the presence of ‘green’ and ‘stem’, which leads the network to correctly conclude the presence of trees and a forest.
In Fig. 2B, no sensory evidence is provided (e.g. the eyes are
closed), but there is a strong prior expectation against the
forest, which leads both belief propagation and circular belief
propagation to correctly predict the absence of trees, leaves,
green and stems.
The fact that circular belief propagation converges to posterior
probabilities on the correct side of 0.5 has been reported previously in the context of circular binary graphs (Frey and McKay,
1998; Murphy et al., 1999). These results predict specifically that
networks with equally impaired upward and downward loops (the
‘equally’ is important, as we will see later) can still infer the most
probable interpretations of their sensory input and priors. Even a
total absence of inhibitory controls does not predict any systematic
bias in perceptual or behavioural decisions as long as these
decisions are based on reliable sensory evidence and/or on
general agreement with prior expectations. The main explanation
for this phenomenon is as follows: while top-down and bottomup messages are indeed counted too many times, the number
of over-counted messages is equal in both directions, leaving
the relative weighting of the sensory evidence and priors intact.
| Brain 2013: 136; 3227–3241
iterations of the network equations (blue circles: physiological network; red lines: impaired network with c = 0.1, d = 0.1). (A) If sensory evidence is entered as an Log-Odd Ratio (LOR)
of 1 for « green » and an LOR of 2 for « stem », then the network converges toward the correct percepts (i.e. the beliefs are on the same side of 0.5), but the confidence is overestimated.
(B) The same as in A, but with no sensory evidence and a prior belief that corresponds to an LOR for a forest of 3 (i.e. a strong prior belief against the forest). (C) If the network receives
very weak sensory evidence for green and stem, then the final belief corresponds to technically correct percepts (on the correct side of 0.5) but with an excessively high confidence for such
weak sensory evidence. (D) Finally, when the network receives contradictory priors and sensory evidence, i.e. the prior is against a forest (LOR of 2), but when sensory evidence suggests
the presence of green and stems (LORs of 1), intermediate representations become frustrated and oscillate between prior-based and sensory-based beliefs, specifically for « tree » and
« root ».
Figure 2 Results for normal and circular inference when inhibitory loops are equally impaired. In each panel, the computed beliefs for all of the nodes x1 to x6 are plotted for different
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R. Jardri and S. Denève
Circular inferences in schizophrenia
Over-confidence
Even if the LOR’s sign is correct, the levels of confidence that are
reached by the impaired and ‘normal’ networks, i.e. the amplitude
of the LOR, are widely different in the circular belief propagation
and belief propagation networks. Because circular belief propagation accounts for sensory and prior evidence not once, but multiple times, it generates levels of confidence that are out of
proportion when compared with the available sensory and prior
information (Fig. 1D). In the example shown in Fig. 2C, the sensory evidence provided to the network is extremely weak. Such
small likelihoods could be the result of: (i) processing extremely
unreliable sensory inputs; (ii) completely ambiguous sensory
inputs; or even (iii) a pure background sensory noise.
Weak sensory evidence can be partly processed and analysed
(e.g. we can sometimes see faces in the random shapes of clouds),
but the confidence that is associated with these interpretations
should be very low. Indeed, the unimpaired network correctly
maintains LORs close to zero in these situations, which indicates
the absence of any reliable conclusions. If perceptual decisions
(and any other forms of decisions) are based on the confidence
in these interpretations, then the system should not perceive trees
or forest when the evidence is too weak. In contrast, the circular
belief propagation network ‘jumps to conclusions’ and reaches a
high level of confidence even when the sensory evidence and/or
the prior are extremely weak. In fact, the circular belief propagation network cannot stably maintain the LORs of any variable near
zero but will amplify any small biases until reaching the maximum
level of confidence that is allowed by the strength of the causal
links Wij . In other words, it is forced to derive a highly confident
interpretation from meaningless or non-significant sensory information. The same phenomena are observed in the presence of a
weak prior (not illustrated here). For example, even a small
expectation of being in the forest could lead the system to perceive trees, leaves and the colour green.
Dissociation between ‘low-level’ and
‘high-level’ representations
In addition to being overconfident, the circular belief propagation
network exhibits a striking dissociative effect between low-level
sensory areas and high-level representational areas in cases in
which sensory evidence contradicts prior beliefs (Fig. 2D). For example, this type of situation could occur when we are inside a city
building and see stems and the colour green (e.g. from a potted
plant). This phenomenon goes against prior expectations because
we are not in a forest, but rather a building, where plants are rare.
Normal belief propagation in the unimpaired network correctly
concludes that while the presence of ‘leaves’ is likely, the presence
of a tree is unlikely. Indeed, while ‘tree’ is supported by ‘leaves’, it
is contradicted by the absence of ‘forest’. Top-down and bottomup evidence cancel each other at an intermediate level in the
hierarchy, preventing the propagation of false beliefs up and
down the hierarchy. Only a few iterations for the belief propagation algorithm are sufficient to reach this stable belief state.
Instead of quickly converging to low confidence in intermediate
layers, however, the circular belief propagation algorithm
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alternates (over successive iterations of the circular belief propagation algorithm) between strong beliefs that are based on sensory evidence alone (a consequence of over-counting bottom-up
messages) and strong beliefs based on prior expectations alone (a
consequence of over-counting top-down messages) (Fig. 2D).
Because both belief states are mutually incompatible, this pattern
results in incoherence between ‘high level’ and ‘low level’ sensory
representations. These oscillations are strongest in the intermediate
layers (where prior and sensory evidence should cancel to give an
LOR that is close to zero). At these intermediate stages, the network alternates between one interpretation of the sensory observations and its exact opposite. When the loops are completely
impaired (c ¼ d ¼ 0) and the causal links are strong (p1ij ¼ 1,
p0ij ¼ 0), such oscillations can even be sustained indefinitely,
which prevents the network from converging to any stable conclusions. When loops are only partially impaired (1 > c ¼ d > 0)
and causal links are less extreme, the amplitude of these oscillations is progressively dampened until the network reaches stable
beliefs, but not before many iterations of the circular belief propagation algorithm have occurred.
Impact of a circular inference on
learning causal relationships
Uncontrolled reverberation of messages is as detrimental to learning as it is to inference. For example, if the node ‘tree’ is frequently co-activated with ‘leaf’, the strength of their causal link
should be reinforced, but only if the activation of ‘tree’ is not
solely due to processing information from ‘leaf’ and comes from
other (non-redundant) sources of information, such as the presence of ‘forest’ or the colour ‘green’ or ‘stem’ (Fig. 1). Otherwise,
the causal link would be reinforced indefinitely, even in the
absence of any true causality, simply because any pre-existing
connection between ‘leaf’ and ‘tree’ will introduce spurious correlations between their associated beliefs, leading to a reinforcement of this link and thus leading to more correlations.
Even when the causal model is accurate, as in the examples
above, circular inference causes over-confidence and dissociative
beliefs on a trial-to-trial basis. However, on a more permanent
(and most likely disruptive) basis, it also results in the learning of
delusional causal relationships, as illustrated in Fig. 3. Figure 3A
represents the learned conditional probability estimate p123 as a
function of the number of training examples, in a situation of
¼ p1true
¼ 0:7,
high sensory and prior uncertainty (i.e. p1true
12
34
0true
0true
p12 ¼ p34 ¼ 0.3). Let us assume that ‘green’ and ‘leaf’ are in
fact completely independent events in the real world, i.e.
¼ p0true
¼ 0:5. This construct could correspond to the imp1true
23
23
aginary case of an observer living in a world where leaves are
not carried by trees. The intact belief propagation network accurately learns that the two variables are not causally linked, i.e. p123
converges to a value close to 0.5. In contrast, the estimate of the
circular belief propagation network (c ¼ d ¼ 0:1) does not stay
close to 0.5; instead, it diverges to a value that is close to 0 or 1,
which implies that the model gradually learns a strong but completely ‘delusional’ (because it does not exist in the real world)
causal relationship between ‘green’ and ‘leaf’.
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R. Jardri and S. Denève
Figure 3 Learning in circular and intact belief propagation networks using the expectation-maximization (E-M) algorithm.
(A) Representation of the learned conditional probability estimates as a function of the number of trials in a situation of high sensory and
prior uncertainty. Although the belief propagation network quickly converges to a value that is close to 0.5 (green), the circular belief
propagation network gradually diverges to 0 or 1 (red), which models how such an impaired system could aberrantly detect coincidences
that do not rely on observable causal relationships. Note that the different curves were obtained by running the E-M algorithm on a
different random sequence of training trials, with different initial values for the parameters. (B) Learned conditional probabilities between
uncorrelated causes as a function of the number of trials, when sensory and prior evidence is more accurate. (C) Learned conditional
probabilities between truly correlated causes as a function of the number of trials in a situation of high sensory and prior uncertainty.
With respect to overconfidence, this problem arises only in situations that have a high level of uncertainty, i.e. when sensory and
prior evidence are relatively unreliable and between variables that
are completely uncorrelated. If priors and likelihoods are strong
and unambiguous, or if the causal relationship is real (i.e. most
of the time), both the intact and circular networks learn accurate causal links. For example, by using more accurate evidence
¼ p1true
¼ 0:9, p0true
¼ p0true
¼ 0.1, while
in Fig. 3c (p1true
12
34
12
34
¼ p0true
= 0.5), we found that the circular belief propagation
p1true
23
34
network estimates converged to the true estimate of 0.5, inferring
that ‘tree’ is unrelated to ‘leaf’. Similarly, even when sensory and
¼ p1true
¼ 0:7, p0true
¼ p0true
¼
prior evidence are weak (p1true
12
34
12
34
0.3), both intact and circular networks can accurately learn
¼ 0:8, p0true
¼ 0:2
a non-random causal relationship p1true
23
23
(Fig. 3B). These examples illustrate how the acquisition of delusional causal relationships is observable only between ‘highly uncertain’ variables that are truly unrelated, and at the same time,
are not strongly corroborated by direct sensory evidence. For example, a tingling sensation in the arm is only weak evidence for
the presence of a micro-chip and, in principle, is not related to the
existence of ‘invisible’ aliens; nevertheless, a patient suffering from
schizophrenia can strongly believe that aliens (that he never directly observed) implanted a micro-chip (that he did not really feel)
in his arm.
Asymmetric impairments of upward and
downward loops
Because upward and downward inhibitory loops most likely rely
on distinct sub-populations of interneurons and/or cortico-basal or
cortico-thalamic loops (see ‘Discussion’ section), both types of
loops could be differentially affected in schizophrenia. Here, we
consider the consequence (for inference) when upward loops are
altered while downward loops remain relatively intact, or vice
versa. Figure 4 illustrates the differential effects of disturbing
mainly upward loops (Fig. 4A, black) or mainly downward loops
(Fig. 4B, green). As described previously, both types of dysfunctions lead to overconfidence (Fig. 4C). Similarly, contradictions
between sensory evidence and priors lead to oscillations in intermediate layers and dissociations between high-level and low-level
representations, albeit to a lesser extent (Fig. 4D and 4E). In addition, however, the two types of impairments affect the relative
weighting of the sensory evidence and priors in strikingly different
fashions.
When upward loops are affected, but downward loops are not,
sensory evidence that is sent up the hierarchy is reverberated back
down (Fig. 4A). As a result, high-level interpretations of the current sensory information are mistaken for prior knowledge.
However, prior knowledge sent down the hierarchy is not reverberated back up, i.e. the system does not mistake its own expectations for sensory evidence (Fig. 4A). As a result, sensory evidence
accumulates over successive layers of the hierarchy and becomes
severely over-counted. However, priors (descending messages) are
processed normally because the downward loops are intact.
Because they are not reverberated, prior evidence is not overcounted. This scenario introduces an effective imbalance in the
relative weighting of sensory evidence and priors. The network
with impaired upward loops relies more on its current sensory
observation than its prior knowledge or its past experience
(Fig. 4D and E).
In contrast, when downward loops are affected, but upward
loops are not, prior expectations that were sent down the hierarchy are reverberated back as if they were additional sensory
evidence (Fig. 4B). Thus, the system mistakes its own prior expectations for new (supporting) sensory observations. In contrast,
sensory evidence that is sent up the hierarchy is not reverberated
back down, i.e. the system does not mistake sensory evidence for
prior expectations (Fig. 4B). As a result, predictions based on priors
are over-counted and accumulate over successive layers. In contrast, sensory evidence is not reverberated and is counted only
once. As a consequence, the network with impaired downward
is sent up in the hierarchy is reverberated back as if it were additional prior evidence (magenta arrows). However, prior evidence sent down the hierarchy is not reverberated back up, i.e.
the system does not mistake its own expectations for sensory evidence (violet arrow). As a result, sensory evidence is overcounted, but not priors. (B) When downward loops are impaired,
but not upward loops, the prior expectation sent down the hierarchy is reverberated back as if it were additional sensory evidence (violet arrows). However, sensory evidence sent up the
hierarchy is not reverberated back down, i.e. the system does not mistake its sensory evidence as prior expectations (magenta arrow). As a result, prior expectations are overcounted, but
not sensory evidence. (C) Impaired networks in the presence of weak sensory evidence (in this example, against the presence of green and stem). Blue: intact network. Green: networks
with impaired downward loops, i.e. c = 0.9, d = 0.1 (as illustrated in B). Black: networks with impaired upward loops, i.e. c = 0.1, d = 0.9 (as illustrated in A). Both dysfunctions result in
overconfidence, i.e. weak sensory evidence leads to highly confident beliefs. (D) Impaired networks in the presence of contradicting prior information (for the presence of forest) and
sensory evidence (against the presence of green and stem). The network with impaired upward loops bases its highly over-confident conclusions entirely on the sensory evidence.
The network with impaired downward loops, in contrast, bases its final beliefs entirely on the prior information. The two networks reach such opposite conclusions for the low-level
variables, even when a normal network typically concludes that the states of these variables are highly uncertain. (E) The same as in D, but for a different combination of likelihoods
and priors.
Figure 4 Results of circular inference with an imbalance between upward and downward loops. (A) When upward loops are impaired but downward loops are not, sensory evidence that
Circular inferences in schizophrenia
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loops relies more on its priors that on its new sensory evidence
when making perceptual decisions (Fig. 4C and D).
We thus can say that, quite paradoxically, an imbalance between upward and downward loops (Fig. 4) has an even more
drastic and disruptive influence on perception and belief formation
than when both loops are equally impaired (Fig. 2). In contrast to
the case of equally impaired loops, a selective impairment creates
an imbalance and results in ‘false percepts’, i.e. in subjects believing in interpretations that are opposite to the results of normal
inference (LORs are of the wrong sign).
Let us finally consider the case of perceptual illusions. In a
normal inference (belief propagation), a strong prior (i.e. slow motions are more likely than fast motions) combined with weak sensory likelihood (i.e. a visual motion stimulus at a low contrast)
results in a systematic bias in the direction of the most a priori
probable interpretations (i.e. the motion of a low contrast visual
stimulus is perceived as slower than it actually is because slower
motions are more likely). This effect is gradual, i.e. the weaker the
sensory likelihood, the stronger the effect of the prior. This phenomenon accounts specifically for many perceptual illusions (Yuille
and Bülthoff, 1996; Weiss et al., 2002; Bogadhi et al., 2011).
Figure 4D and E show two examples in which weak sensory evidence is combined with a strong prior. The circular belief propagation framework predicts a strong disruption of the normal
perceptual biases when the upward and downward loops are differentially impaired. Whereas a selective impairment of the downward loops predicts dominance of the prior and thus a stronger
effect of perceptual illusions, a selective impairment of the upward
loops results paradoxically in fewer perceptual illusions than are
normally expected. Thus, assessing the relative weighting of sensory observations and priors is an indirect way to assess whether
upward or downward loops are primarily impaired (or vice versa).
Discussion
The aim of the current paper was to introduce the concept of
circular belief propagation in causal networks as a result of an
excitatory to inhibitory imbalance in hierarchical cortical networks.
We now discuss more conceptually how it could account not only
for inappropriate causal attributions, bizarre beliefs and hallucinations in the schizophrenia spectrum but also for emerging aberrant coincidence detections during the transition phase to
psychosis.
The ‘jumping to conclusions’
phenomena
The most salient effect of circular belief propagation is to reach
levels of confidence that are out of proportion with the true levels
of uncertainty that are associated with the sensory data and/or
prior knowledge. Such over-interpretation of weak sensory observations could account for the ‘jumping to conclusions’ bias that is
observed in schizophrenia patients (Huq et al., 1988). The jumping
to conclusions phenomenon is traditionally described using probabilistic reasoning tasks such as the ‘beads task’. Two jars are
presented to the participant with different proportions of ‘red’
R. Jardri and S. Denève
and ‘black’ beads, and the task is to successively sample beads
from one of the two jars. The subject must decide from which jar
the beads are taken (based on the proportion of beads in each
jar). Using such a paradigm, a number of behavioural studies have
showed that patients with paranoid schizophrenia, when faced
with probabilistic choices, base their decision on far less evidence
while reporting much higher confidence than healthy controls.
Crucially, the jumping to conclusions phenomenon seems to be
specifically associated with delusional tendencies (Huq et al.,
1988; Garety et al., 1991; Moritz and Woodward, 2005;
Speechley et al., 2010; Averbeck et al., 2011).
Importantly, the circular belief propagation framework can be
distinguished from Bayesian models assuming degraded sensory
information or the use of suboptimal decision criteria (Averbeck
et al., 2011; Moutoussis et al., 2011), but considering that the
inference itself is not affected (Fear and Healy, 1997). In line with
an impaired inference process, a recent experiment confirmed that
when explicitly required to report their confidence level in a variant of the ‘beads task’, patients with schizophrenia systematically
overestimated the weight of the sensory evidence (Speechley
et al., 2010).
Uncertainty, hallucinatory and
delusional experiences
Interestingly, the dysfunctions observed in the circular belief
propagation models were not found to be perpetual in the sense
that the system was able to interpret appropriately the stimuli
most of the time. Erroneous beliefs [e.g. jumping to conclusions
(phenomenon)] were shown to emerge only when the system was
facing ambiguous or contradictory information, i.e. when the
probabilities were close to 0.5. Such situations are presumably
relatively rare in everyday life, where our senses and priors provide
us (most of the time) with strong and unambiguous information.
The low frequency of ambiguous situations could account for the
very intermittent nature of hallucinatory experiences (Jardri et al.,
2013) but could also explain why hallucinations are more frequent
when there is sensory deprivation (Zubek, 1964). A lack of sensory
input could, in this situation, put causal networks into a state of
high uncertainty, leading to an increased occurrence of erroneous
percepts. Furthermore, this scenario could account for the frequent persecutory nature of paranoid delusions because social
inferences are tainted with much more uncertainty than simple
perceptual inferences in which the relationship between the
cause and the evidence is more straightforward (Chambon
et al., 2011).
Learning of delusional causal models
We showed that circular belief propagation disrupts the control of
messages and results in the reverberation of top-down and
bottom-up evidence, which occurs multiple times. Thus, even in
the absence of any true causal relationships in the outside world,
top-down and bottom-up messages can be strongly correlated
(e.g. the messages are mixtures of violet and magenta arrows,
in Fig. 4). As a result, inaccurate causal links can be progressively
and recursively strengthened, which leads to highly trusted causal
Circular inferences in schizophrenia
models that are not corroborated by sensory experience. This dynamic phenomenon fits especially well with the abnormal experiences that are described in the initial phase of psychotic
experiences (Woods et al., 2009). During this prodromal period,
patients describe the appearance of mild or brief non-specific
symptoms, such as a noticeable feeling of strangeness or aberrant
coincidence detections. Interestingly, a low dose administration in
healthy volunteers of ketamine, an NMDA receptor antagonist
drug, also appears to mimic prodromal states rather than the full
picture of schizophrenia (Pomarol-Clotet et al., 2006). Such a
drug-induced prepsychotic state supports the progressive learning
of aberrant causal associations that has been demonstrated by
circular belief propagation models.
Beyond the emergence of delusional ideas, circular belief propagation models also account for their remarkable persistence.
Indeed, an insufficient compensation for internally generated correlations by inhibitory loops could lead to the consolidation of
extremely strong delusional causal models based on weak or insignificant sensory evidence or prior beliefs, thus forcing paranoid
interpretations (though not entirely illogical) of random events or
meaningless coincidences. This finding is clearly confirmed in the
clinical setting, in which psychotic patients were shown to specifically over-emphasize evidence that inflates the veracity of their
delusional beliefs but not evidence that could disprove it (Garrett
and Singh, 2012; Kaliuzhna et al., 2012). This phenomenon
accounts for the abnormal maintenance, over time, of beliefs
that cannot be criticized (Corlett et al., 2009b).
Selective impairment of upward loops in
schizophrenia?
The circular belief propagation framework distinguishes between
different classes of circular inference by considering bottom-up
and top-down streams separately. We have seen that impaired
upward loops result in sensory evidence that is reverberated as if
it were a prior expectation, which causes sensory evidence to be
over-counted. The opposite also occurs: impaired downward loops
result in prior expectations being reverberated as if they were
sensory evidence, which causes priors to be over-counted. These
two forms of impairment could both be implicated in the schizophrenia spectrum, which would account in part for the large heterogeneities that are observed in patients’ behaviours (Tandon
et al., 2009).
It has been previously proposed in the literature that positive
symptoms could result from an underestimation or impairment of
prediction errors, which would lead patients with hallucinations
and delusions to rely more on their priors (Grossberg, 2000;
Friston, 2005a; Fletcher and Frith, 2009). Such a hypothesis is in
line with an impairment of downward loops in the current framework. However, an over-counting of the prior does not appear to
be fully compatible with the spectrum of behaviour that is
observed in schizophrenia. For example, let us return to the jumping to conclusions phenomena that were previously described. In
the beads task, patients with delusional tendencies base their decision on far fewer samples (often only one sample suffices) while
reporting much higher confidence compared with healthy control
Brain 2013: 136; 3227–3241
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subjects. If these patients under-estimate the reliability of new
sensory observations (i.e. a newly sampled ‘black’ bead) compared
to what they already know (i.e. the probability of each urn based
on beads that were observed previously), they should, in contrast,
give a smaller weight to each new sample, thus accumulating
evidence and confidence more slowly than healthy controls.
Jumping to conclusions suggests, in contrast, that patients overinterpret their sensory evidence compared to their prior knowledge, which could be the result of impaired upward loops.
In additional support of this hypothesis, patients with schizophrenia are paradoxically less vulnerable to illusory percepts than
healthy control subjects (Dakin et al., 2005; Tschacher et al.,
2006; Crawford et al., 2010; Williams et al., 2010). This attenuation specifically appears in patients with schizophrenia prone to
positive symptoms (Shergill et al., 2005), but may also be
observed in healthy individuals that score highly on delusional beliefs (Teufel et al., 2010). Knowing that the biases that were
introduced by prior beliefs are considered to be at the root of
perceptual illusions (Geisler and Kersten, 2002; Weiss et al.,
2002), these findings support an over-estimation of the strength
of sensory evidence and an underweighting of the prior, which is
compatible with an impairment of the upward loops.
Based on these converging data, we thus propose that hallucinations and delusions originate primarily from the reverberation
of sensory evidence as top-down expectations from high-level
areas to low-level areas, leading to an over-interpretation of
the sensory evidence. This, however, does not rule out the possibility that some patients are also impaired in their downward
loops, and thus, over-count their priors (Friston, 2008; Chambon
et al., 2011). Importantly, these two hypotheses could be tested
experimentally by measuring how patients weight their likelihood
and priors during decisions. Finally, an impairment of downward
loops (i.e. the reverberation of top-down expectations as if they
were sensory evidence) could cause vivid mental imagery, as
observed, for example, during normal human development
before complete frontal GABAergic maturation (Uhlhaas et al.,
2010).
We would like to highlight that, in the circular belief propagation framework, positive symptoms are understood in terms of
false inferences in a causal hierarchy and not as the consequence
of focal brain damage (the previously described diffuse inhibitory
loops impairment is clearly different from what could be expected
from a neural lesion). As shown in Supplementary Fig. 3, low-level
damage (e.g. at the level of the retina or of the pedonculus) can
generate erroneous sensory inputs (Supplementary Fig. 3A), but
because inferences are correctly processed during belief propagation, the subject continues to be able to challenge and disprove
these percepts. Such a preserved insight is effectively a prominent
feature of hallucinosis (Ey, 1973; Blom, 2010), compared with the
psychotic hallucinations that are observed in schizophrenia. In contrast, when lesions are located in high-level brain structures
(e.g. at the level of the pre-frontal cortex), incorrect priors can
be generated, which results in common symptoms of the dysexecutive syndrome, such as perseverations; however, these patients
do not suffer from abnormal sensory judgements (Supplementary
Fig. 3B).
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The relationship between circular belief
propagation and gain control
False inference has now become a dominant theme in the study of
the positive symptoms (reality distortion) of schizophrenia and
related syndromes. Generally, this is cast in terms of abnormal
representations of uncertainty or—more specifically—estimates of
the precision of sensory evidence relative to top-down prior beliefs
(Friston, 2005a; Fletcher and Frith, 2009). In the Bayesian formalism, the precision of sensory evidence and prior knowledge determines their gain, i.e. their relative contributions. Interestingly, one
of the main consequences of circular belief propagation is the
resulting change in the confidence assigned to sensory evidence
and top-down priors. This is mediated by multiple reverberations
of the same redundant message, and will worsen with the number
of layers and the number of iterations of the algorithm. As a
result, the ‘normal’ gain of the sensory evidence and/or prior drastically increase as c and d decrease from 1 (normal inhibition) to
zero (no inhibition).
The relationship with predictive coding
In this paper, we focus on simple models with discrete states that
do not change with time. In the case of continuous time-varying
variables, belief propagation can be efficiently approximated (or in
some cases exactly implemented) by a form of generalized predictive coding. As in belief propagation, the generalized predictive
coding framework has been used to interpret hallucinations and
delusions as an impairment in the neural representation of uncertainty (Friston, 2005a; Adams et al., 2013). In addition, belief
propagation and generalized predictive coding are algorithmically
similar. Both belief propagation and generalized predictive coding
are minimizing an energy function (Friston, 2005b; Friston et al.,
2010; Balaji and Friston, 2011). As for belief propagation, generalized predictive coding estimates the precision (gain) of bottom-up
and top-down evidence online in order to combine them properly.
Finally, generalized predictive coding updates internal representations according to the error between the bottom-up sensory
evidence and its top-down predictions (i.e. the prediction error).
This mechanism is analogous to the downward loops in belief
propagation, in which ‘bottom-up’ messages are corrected by
the previous redundant ‘top-down’ message (and vice versa).
Although a systematic and rigorous comparison of two theoretical
approaches extends beyond the scope of this paper, these similarities clearly imply that the two frameworks are related and
mutually compatible.
Possible neural substrates
Different models have been proposed for the specific neural implementation of belief propagation in cortical circuits (Lee and
Mumford, 2003; Deneve, 2004; Friston, 2005b; Bishop, 2006;
George and Hawkins, 2009; Steimer et al., 2009; Markov and
Kennedy, 2013). Importantly, regardless of its specific implementation, belief propagation relies on local corrective loops to avoid
circular inference, a property shared with ‘predictive coding’ (see
above). The presence of these corrective loops has been shown to
R. Jardri and S. Denève
be highly compatible with the architecture, connectivity and
dynamics of the cortical column (Bastos et al., 2012).
In addition, the notion of separate feed-forward and feedback
inhibitory pathways (implementing on upward and downward
loops, respectively) builds on a series of anatomical and physiological arguments. It can—at least roughly—be motivated by the
distinction between superficial and deeper pyramidal cells that
are the source of forward and backward connections, respectively
(Coogan and Burkhalter, 1993; Markov et al., 2011). Furthermore,
recent electrophysiological studies suggest that feed-forward connections rely on higher synchronization frequencies than feedback
connections (Bosman et al., 2012). Another potential implementation for these inhibitory pathways relies on the mediation or
control of cortical communications by subcortical structures. For
example, the ventral striatum was proposed as a key structure in
the pathophysiology of positive symptoms. Abnormally salient
experiences in schizophrenia could relate to an increased tone
within the mesolimbic dopamine system (Kapur, 2003), whereas
D2 receptor blockage by antipsychotic medication was shown to
be critical to symptom resolution (Creese et al., 1976; Johnstone
et al., 1978). Moreover, increased activity within the mesolimbic
system correlates with the severity of hallucinatory experiences
(Howes et al., 2007).
Accounting for other clinical dimensions
of schizophrenia?
Circular belief propagation provides a new framework for interpreting the generation of unusual sensory experiences or strange
beliefs, which progressively become unshakable. From a macroanatomical view, we can conclude that circular belief propagation
models with impaired inhibitory loops are fully compatible with
dysconnectivity hypotheses that are supported by several imaging
findings in schizophrenia and, more specifically, in hallucinations
(Stephan et al., 2009; de Weijer et al., 2011; Amad et al., in
press). Indeed, circular belief propagation acts by disrupting communications between brain areas. However, does circular belief
propagation account for the more rarely modelled dimensions of
schizophrenia, such as cognitive disorganization or negative
symptoms?
When upward inhibitory loops are selectively impaired, the
system could be forced to take into account ‘overweighed’ sensory
information even if such information is unreliable. This possibility
perfectly matches with experimental findings from the classical
latent inhibition paradigm, which measures the ability to filter
irrelevant stimuli, a capability that is found to be decreased in
patients with schizophrenia (Lubow and Weiner, 2010). This proposal also fits with the high distractedness and attentional impairments that are observed in schizophrenia (Tandon et al., 2009).
Finally, the emergence of frustrated states in circular belief
propagation networks provides a potential mechanism for the occurrence of concomitant ambivalent ideas in schizophrenia patients
who suffer from dissociation (Fig. 2D). We effectively showed that
circular belief propagation predicts incoherence and instability in
the beliefs that were generated by intermediate levels of representations whenever sensory evidence and prior contradicted each
Circular inferences in schizophrenia
other. In that situation, high-level and low-level representations
became effectively disconnected, i.e. the network derived highly
confident low-level and high-level interpretations that were mutually incompatible; in extreme cases, it believed simultaneously in
two opposite interpretations of the same sensory evidence. Such a
dissociation between higher and lower level inferential processes
nicely relates to the clinical concepts of ‘double orientation to
reality’ by Jaspers (1963) or ‘double entry bookkeeping’ by
Bleuler (1950), which both refer to the fact that patients with
schizophrenia might live with two separate and conflicting sets
of beliefs.
Conclusions
Within the broader field of psychiatry, a background argument has
been to determine whether psychosis was essentially characterized
by anomalous perception leading to incorrect (but essentially rational) inferences or, on the contrary, may rely on normal perceptions but irrational inferences. Crucially, as suggested by Fletcher
and Frith (2009), the Bayesian framework renders the argument
itself unimportant because a single deficit acting at multiple levels
may account for both perceptual and belief/inferential anomalies
in psychosis. In line with this approach, we used an analogy
between hierarchical neural processing and belief propagation to
illustrate how excitatory to inhibitory imbalances might result in
circular inference. Our framework is normative, highly testable,
and quantitative, regardless of the specific neural implementation
of the ‘inhibitory loops’. One way of testing the circular belief
propagation framework would be, for example, to assess carefully
the weight that is associated with sensory evidence and priors in
perceptual decisions. Beyond the theoretical proofs of concepts
that were provided here, further developments are now required
to improve the biological plausibility of circular belief propagation
networks, to explore the possible neural correlates of inhibitory
loops and to provide a better translational understanding of hallucinations and delusions.
Acknowledgements
We would like to thank Philippe Domenech, Izzet Burak Yildiz and
Delphine Pins for reading and discussing a preliminary version of
this work. We also thank the reviewers for their help in improving
the manuscript.
Funding
RJ was partially supported by the Pierre Houriez Foundation
(hosted by the Fondation de France), grant no. FdF-200462/
2009-02.
Supplementary material
Supplementary material is available at Brain online.
Brain 2013: 136; 3227–3241
| 3239
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