Figure 1.
(A) Anatomical reconstruction of the dendritic tree of a mature granule cell from mice. The dendritic tree is dominated by the long parallel dendritic branches in the outer two third of the molecular layer. Note the lack of basal dendrites and the apical trunk compared to a pyramidal neuron. Image courtesy of Dr. Josef Bischofberger. (B) Model for the somato-dendritic interactions in dentate granule cells. Distal dendritic compartments are represented by circles, and the soma by a square. Further details are in the text. (C) The different dendritic integration functions used in this study. Black: linear, blue: quadratic, green: sigmoid function. Red, square symbols indicate the nonlinearity of dendritic integration in a conductance based model of hippocampal granule cell (see Text S1). Dashed and dotted lines show different degrees of nonlinearity. The distribution of the total dendritic input with uniform synapses is shown in the background.
Figure 2.
Input strength encoding with uniform synapses.
(A) The figure shows two neurons (or the same neuron with two different input sets). We calculated the probability of firing (H(U)) given that one of the branches has exactly U synaptic input (e.g., U1 = U) while inputs of other branches (U2, U3, …, UN) are drawn independently from the input distribution. Second, we calculated the distribution of the maximal input (K(U*)) given the depolarization of the soma exceeds the firing threshold. (B) Color coded joint probability distribution of the somatic activation and the maximal dendritic input, with the linear integration function. Red is maximum, dark blue is zero. The color-code emphasizes low probability events and it is not linear. The horizontal line is the firing threshold; the yellow line shows the conditional expectation of as given U*. If dendrites were independent high and low U* values could be separated by a somatic threshold of action potential generation. (C–D) Dendritic independence with linear (C) and quadratic (D) integration functions. Left axis, red: K(U*), the distribution of the maximal dendritic inputs during firing. Right axis, blue: the H(U) function, which is the probability of firing given that one of the dendrites has U total input. The probability of triggering output by a single branch is low (H(U)<0.25) even with reasonably large input (as revealed by the low H(U) values at the probability mass of K(U*)). This indicates that a single dendritic subunit is unable to reliably activate the neuron with these integration functions. Background light gray is the distribution of U while dark grey shows the distribution of U*. Parameters: R = 0.01, N = 30.
Figure 3.
Independent feature detection with Habbian synapses.
(A) Synaptic plasticity separates the inputs. Before learning the total synaptic input to a dendritic subunit come from a Gaussian distribution (600 samples are shown with grey circles) with the calculated density function shown on the right (Eq. 5). During the learning process each branch learns its largest input and the response increases to the learned input (blue circle), while it decreases to all other inputs (black circles). The black and blue Gaussian curves show the density functions for the non-learned and learned inputs, respectively (Eq. 19). (B), (D): Color-coded joint distribution of the somatic activation and the maximal dendritic input () in the linear (B) and quadratic (D) case. The horizontal lines indicate the firing threshold. (C), (E): The distribution of the maximal dendritic input when the cell fires (K(U*) in red) and the probability of firing with a given input (H(U) in blue). The distribution of U* is shown in the background. In the linear case, 50% of firing occurs when one of the branches receives its preferred input, while with quadratic integration function more than 95%. Parameters: R = 0.01, N = 30.
Figure 4.
Changing the resistance influence the isolation of compartments, but not the detection of dendritic spikes.
(A) The joint distribution of the somatic activation as and the maximal dendritic input U* with different resistances. (B) The probability of detecting a dendritic spike remains constant even if the resistance changes 2–3 order of magnitude. Black circles: both somatic and dendritic resistances are altered; green diamonds: only resistance of the dendritic membrane () is changed while
; red squares: somatic membrane resistance (
) is changed,
. Although the distribution of the somatic activation scales with the resistance (see panel A and Eq. 7), the detection probability of a single dendritic event remains relatively constant. (C) The joint distribution of the activation of a dendritic branch
and its own input Ui. When the resistance is low (left), the local activation depends only slightly on the input. Conversely, if the resistance is higher (right) the local input has substantial impact on the activation of the branch. (D) The external influence decreases as the resistance increases. Symbols are the same as on panel B. Note, that decreasing the resistance of the perisomatic membrane (red squares) is the most efficient in separating the dendritic compartments (recall, that
). N = 30.
Figure 5.
Moderate number of branches allows the isolation of subunits and the detection of dendritic spikes.
(A) The joint distribution of the somatic activation as and the maximal dendritic input U*. As the number of branches grow (from left to right), the somatic depolarization caused by a dendritic spike in a single compartment decreases gradually. Consequently, if N is high, than the somatic threshold (horizontal line) can not separate small and large dendritic events. (B) The probability of detecting a dendritic spike decreases as the number of dendritic subunits increases. Different colors indicate different degrees of nonlinearity (blue circles: quadratic; black triangles: linear integration function). (C) The joint distribution of the activation of a dendritic branch and its own input Ui. Increasing the number of compartments decrease the variance of the distribution and the impact of other branches. (D) The external influence decreases with the number of branches both with linear (triangles) and quadratic (circles) integration function. R = 0.3.
Figure 6.
Location dependent input and parallel dendritic computations generate multiple place fields.
The behavior of the same granule cell in five different environments (columns). Upper row: color-coded maps (“ratemaps”) show the somatic activation on the 1×1 meter large maze. Red: high activation (spiking), blue: silent. The highest and the lowest value of the somatic activation is indicated on each ratemap. The places where the activation exceed the threshold (“place fields”) are surrounded by black lines. We used the same, linear color-code in all panels. Lower row: the track of the robot and the location of the dendritic spikes. Dendritic place fields of different branches of the same neuron are marked by different colors. Somatic firing usually coincide with the activation of single dendritic branches.
Figure 7.
Spatial firing patterns with different number of dendritic subunits.
(A) The number of dendritic place fields increases with the number of subunits although the probability of learning were kept constant. (B) The correlation between the somatic activity (as) and the maximal input (U*) decreases if the number of dendritic subunits increases. Open circles: correlation in the absence of dendritic spikes. Error bars on A and B show the standard deviation of 50 trials in 5 different environment. (C–D) Spatial firing patterns with N = 4 (C) and N = 100 (D) dendritic subunits. Upper and middle row is the same as on Figure 6. Lower row: scatter plot showing the joint distribution of the somatic activation as and the maximal dendritic input U*. Threshold for synaptic plasticity in the branches (horizontal line) and somatic firing (vertical line) are indicated. The correlation between the two variables is shown above the panels. If the neuron has a small number of dendritic subunits than the number of dendritic fields is small (A, C) but they propagate efficiently to the soma. Conversely, a neuron with a large number of dendritic subunits might have numerous dendritic fields, but the individual dendritic spikes have a little impact on the somatic activation.
Table 1.
Membrane parameters of hippocampal granule cells.