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. 2019 Mar 6;15(3):e1006757.
doi: 10.1371/journal.pcbi.1006757. eCollection 2019 Mar.

Tuft dendrites of pyramidal neurons operate as feedback-modulated functional subunits

Florian Eberhardt  1   2 , Andreas V M Herz  1   2 , Stefan Häusler  1   2
Affiliations

Tuft dendrites of pyramidal neurons operate as feedback-modulated functional subunits

Florian Eberhardt et al. PLoS Comput Biol. .

Abstract

Dendrites of pyramidal cells exhibit complex morphologies and contain a variety of ionic conductances, which generate non-trivial integrative properties. Basal and proximal apical dendrites have been shown to function as independent computational subunits within a two-layer feedforward processing scheme. The outputs of the subunits are linearly summed and passed through a final non-linearity. It is an open question whether this mathematical abstraction can be applied to apical tuft dendrites as well. Using a detailed compartmental model of CA1 pyramidal neurons and a novel theoretical framework based on iso-response methods, we first show that somatic sub-threshold responses to brief synaptic inputs cannot be described by a two-layer feedforward model. Then, we relax the core assumption of subunit independence and introduce non-linear feedback from the output layer to the subunit inputs. We find that additive feedback alone explains the somatic responses to synaptic inputs to most of the branches in the apical tuft. Individual dendritic branches bidirectionally modulate the thresholds of their input-output curves without significantly changing the gains. In contrast to these findings for precisely timed inputs, we show that neuronal computations based on firing rates can be accurately described by purely feedforward two-layer models. Our findings support the view that dendrites of pyramidal neurons possess non-linear analog processing capabilities that critically depend on the location of synaptic inputs. The iso-response framework proposed in this computational study is highly efficient and could be directly applied to biological neurons.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Detailed model of a CA1 pyramidal cell.
(A) Morphology of the cell. The basal and the apical dendrite are shown in gray and black, respectively. Red arrows mark the location of the recording site and two dendritic branches receiving synaptic input. (B) Apical dendrite flattened in 2D. Branches are labeled according to their location in the proximal (P) dendrite or the tuft (T). The first number in a label denotes the subtree index and the second number denotes the branch index. (C) Somatic membrane depolarization for synaptic input at site 1 and three different peak conductance values. The depolarization for the highest peak conductance (purple line) is larger than the sum (dashed line) of the depolarizations for the two lower peak conductances (red and blue line) indicating supralinear dendritic integration. (D) The response r is defined as the membrane voltage averaged over 50 ms. (E) Dependence of the response on the synaptic peak conductance at site 1. The cross indicates the linear sum of the responses for the two lower peak conductances. (F) Dependence of the response on the synaptic peak conductances at site 1 and site 2.
Fig 2
Fig 2. Identification of the feedforward two-layer model with iso-response methods.
(A) Feedforward two-layer model. (B-D) Responses of three models consisting of different supralinear subunit non-linearities (bottom and right insets). Black, purple and yellow lines indicate iso-response curves. Top insets show iso-response curves from above. (E-G) Reconstruction of the first layer subunit non-linearities from the purple and yellow iso-response curves shown in D. (F,G) Black crosses mark reconstructed points on the graph of the subunit non-linearities. The blue and the dark red line show the true subunit non-linearities used to generate the response in D. Note that the axes in G are flipped and Δ denotes a constant step size. (H) The prediction error is measured by the variance of the predicted response on an additional test iso-response curve (not shown) normalized by the variance of the responses used for training (purple and yellow lines).
Fig 3
Fig 3. Performance of point-neuron models and feedforward two-layer models.
(A-C) Prediction error of point-neuron models for pairs of P-P (A), T-T (B) and T-P (C) branches. The first number in a label denotes the subtree index and the second number denotes the branch index. Labels P00-P05 denote the segments of the proximal apical trunk, which are closest to the soma. (D-E) Prediction error of feedforward two-layer models. (G-H) Comparison of both models. (J-L) Normalized subunit non-linearities for two-layer models with low prediction error (< 4%) aligned to the onset of their non-linearity. The normalization involves a vertical and horizontal scaling and shift of each curve.
Fig 4
Fig 4. Two-layer model with feedback.
(A) Feedback modulates the subunit inputs by an additive or multiplicative term (indicated by the placeholder #). These feedback terms are non-linear functions (g1 and g2) of the response. (B-D) Impact of feedback on the response for the two-layer model shown in Fig 3D. (B) Negative feedback shifts the iso-response curves further apart. (C) Positive feedback shifts the iso-response curves closer together. (D) Multiplicative feedback rescales the iso-response curves. Top insets show iso-response curves from above. Dashed lines illustrate the purple iso-response curve of the two-layer model without feedback (Fig 3D). Bottom and right insets show subunit non-linearities and feedback functions. The subunit non-linearities of the two-layer models without feedback are illustrated by the blue and the dark red dashed lines. Solid lines indicate the subunit non-linearities of the corresponding two-layer models with feedback for stimuli on three iso-response curves (black, purple and yellow lines). The feedback is constant on iso-response curves and shifts and rescales the subunit non-linearities. On the gray iso-response curves the feedback is zero.
Fig 5
Fig 5. Performance of two-layer models with feedback.
(A,D) Prediction error of two-layer models with additive feedback for T-T (A) and T-P (D) branch pairs. (B,E) Prediction error of two-layer models with additive and multiplicative feedback. (C,F) Errors of the two-layer models with feedback in units of the errors of the feedforward two-layer models. (G,H) Strength of additive feedback for T-T and T-P branch pairs. (I,J) Strength of multiplicative feedback for T-T and T-P branch pairs. (K) Performance comparison across models.
Fig 6
Fig 6. Examples of performance improvements through feedback.
(A) Response for the branch pair T25-T33 and five iso-response curves (black lines). (B) Error of the feedforward two-layer model in dependence on the input regime. Yellow regions indicate failures of the feedforward two-layer model. The error is given by |log2(tan(αpred)) − log2(tan(αact))|. (C) Two reconstructions of each subunit non-linearity assuming a feedforward two-layer model. The two reconstructions of f1 differ by a horizontal shift. (D) Two reconstructions of each subunit non-linearity assuming a two-layer model with additive feedback. The reconstructed subunit non-linearities are nearly identical. The additive feedback compensates for the shift of f1 in C. (E) Response for the branch pair P14-T25 and five iso-response curves (black lines). (F) Error of the feedforward two-layer model in dependence on the input regime. (G) Two reconstructions of each subunit non-linearity assuming a two-layer model with additive feedback. Each of the reconstructed subunit non-linearities differs by a vertical rescaling. (H) Two reconstructions of each subunit non-linearity assuming a two-layer model with additive and multiplicative feedback. Again, the reconstructed subunit non-linearities are nearly identical. The multiplicative feedback compensates for the rescaling of f1 and f2 in G.
Fig 7
Fig 7. Calcium currents disrupt subunit independence.
(A) Performance of feedforward two-layer models for a representative subset of branch pairs (see S9 Fig). Voltage-gated calcium currents (ICa), an A-type potassium current (IA), and a hyperpolarization-activated cation current (Ih) are blocked individually in the tuft. If voltage-gated calcium currents are blocked, dendritic integration can be explained by a feedforward two-layer model. (B) Performance of point-neuron and two-layer models that predict the somatic firing rate in response to synaptic Poisson input for the same representative subset of branch pairs as used in A. The feedforward two-layer model is sufficient to describe neuronal firing-rate responses even with intact ICa. (C-E) Normalized subunit non-linearities for the feedforward two-layer models in B aligned to the onset of their non-linearity. The normalization involves a vertical and horizontal scaling and shift of each curve.
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Publication types

Grants and funding

The work of FE, AH and SH has been supported by the Federal Ministry of Education and Research through the Bernstein Center for Computational Neuroscience Munich (01GQ1004A, http://www.bccn-munich.de/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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