Is summation linear in a passive membrane?

Is summation linear in a passive membrane?

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I am not too sure what it means for summation to be linear? I am running a simulation and as I decrease time for the second EPSP the amplitude decreases. Does summation being linear mean that there is a direct negative linear relationship between delay of onset and amplitude? I also noticed there is no change when I double or triple my conductance value.

I am just not too sure what the relationship is telling me.

For a membrane to satisfy the conditions for linear summation implies that when two EPSPs of amplitude $A$ and $B$ are summed at the integration zone, the resulting output signal will be of amplitude $A+B$. Since the signal dissipates due to time and length constants, increasing the delay between the two inputs means that the later input will be dissipated so $A+B$ will be smaller than it would have if the delay was shorter. Under this scenario, inputs $A$ and $B$ are still summed linearly. However, if the delay is shorter than the time constants or the inputs are delivered closer in space than the length constant, then you may get non-linear summation since the output signal at the integration zone will be larger than $A+B$.

Have a look here for more information (especially Figure 7):
Silver RA. Neuronal arithmetic. Nature reviews Neuroscience. 2010;11(7):474-489. doi:10.1038/nrn2864.

Somatosensory Perception

Temporal Summation

Temporal summation for vibratory stimuli parallels the findings with spatial stimuli that is, temporal summation is evident with stimulus conditions that favor stimulating Pacinian corpuscles. Higher temporal frequencies and larger contact surfaces yield the greatest amount of temporal summation. With these vibratory stimuli, threshold declines as the duration of stimulation increases out to 500 ms. Based on these and other results, it has been suggested that the Pacinian system (Pacinian receptors and their peripheral and central connections) is, in contrast to the other mechanoreceptor systems, capable of both spatial and temporal summation.

Although not as widely studied as temporal summation for vibratory stimuli, impulse stimuli also demonstrate temporal summation. Increasing the duration of a pulse from 0.35 ms out to 10 ms results in a more than threefold drop in threshold (amplitude of the pulse). Beyond 5 ms, the threshold remains nearly constant.

5.2 Passive Transport

By the end of this section, you will be able to do the following:

  • Explain why and how passive transport occurs
  • Understand the osmosis and diffusion processes
  • Define tonicity and its relevance to passive transport

Plasma membranes must allow certain substances to enter and leave a cell, and prevent some harmful materials from entering and some essential materials from leaving. In other words, plasma membranes are selectively permeable —they allow some substances to pass through, but not others. If they were to lose this selectivity, the cell would no longer be able to sustain itself, and it would be destroyed. Some cells require larger amounts of specific substances. They must have a way of obtaining these materials from extracellular fluids. This may happen passively, as certain materials move back and forth, or the cell may have special mechanisms that facilitate transport. Some materials are so important to a cell that it spends some of its energy, hydrolyzing adenosine triphosphate (ATP), to obtain these materials. Red blood cells use some of their energy doing just that. Most cells spend the majority of their energy to maintain an imbalance of sodium and potassium ions between the cell's interior and exterior, as well as on protein synthesis.

The most direct forms of membrane transport are passive. Passive transport is a naturally occurring phenomenon and does not require the cell to exert any of its energy to accomplish the movement. In passive transport, substances move from an area of higher concentration to an area of lower concentration. A physical space in which there is a single substance concentration range has a concentration gradient .

Selective Permeability

Plasma membranes are asymmetric: the membrane's interior is not identical to its exterior. There is a considerable difference between the array of phospholipids and proteins between the two leaflets that form a membrane. On the membrane's interior, some proteins serve to anchor the membrane to cytoskeleton's fibers. There are peripheral proteins on the membrane's exterior that bind extracellular matrix elements. Carbohydrates, attached to lipids or proteins, are also on the plasma membrane's exterior surface. These carbohydrate complexes help the cell bind required substances in the extracellular fluid. This adds considerably to plasma membrane's selective nature (Figure 5.7).

Recall that plasma membranes are amphiphilic: They have hydrophilic and hydrophobic regions. This characteristic helps move some materials through the membrane and hinders the movement of others. Non-polar and lipid-soluble material with a low molecular weight can easily slip through the membrane's hydrophobic lipid core. Substances such as the fat-soluble vitamins A, D, E, and K readily pass through the plasma membranes in the digestive tract and other tissues. Fat-soluble drugs and hormones also gain easy entry into cells and readily transport themselves into the body’s tissues and organs. Oxygen and carbon dioxide molecules have no charge and pass through membranes by simple diffusion.

Polar substances present problems for the membrane. While some polar molecules connect easily with the cell's outside, they cannot readily pass through the plasma membrane's lipid core. Additionally, while small ions could easily slip through the spaces in the membrane's mosaic, their charge prevents them from doing so. Ions such as sodium, potassium, calcium, and chloride must have special means of penetrating plasma membranes. Simple sugars and amino acids also need the help of various transmembrane proteins (channels) to transport themselves across plasma membranes.


Diffusion is a passive process of transport. A single substance moves from a high concentration to a low concentration area until the concentration is equal across a space. You are familiar with diffusion of substances through the air. For example, think about someone opening a bottle of ammonia in a room filled with people. The ammonia gas is at its highest concentration in the bottle. Its lowest concentration is at the room's edges. The ammonia vapor will diffuse, or spread away, from the bottle, and gradually, increasingly more people will smell the ammonia as it spreads. Materials move within the cell’s cytosol by diffusion, and certain materials move through the plasma membrane by diffusion (Figure 5.8). Diffusion expends no energy. On the contrary, concentration gradients are a form of potential energy, which dissipates as the gradient is eliminated.

Each separate substance in a medium, such as the extracellular fluid, has its own concentration gradient, independent of other materials' concentration gradients. In addition, each substance will diffuse according to that gradient. Within a system, there will be different diffusion rates of various substances in the medium.

Factors That Affect Diffusion

Molecules move constantly in a random manner, at a rate that depends on their mass, their environment, and the amount of thermal energy they possess, which in turn is a function of temperature. This movement accounts for molecule diffusion through whatever medium in which they are localized. A substance moves into any space available to it until it evenly distributes itself throughout. After a substance has diffused completely through a space, removing its concentration gradient, molecules will still move around in the space, but there will be no net movement of the number of molecules from one area to another. We call this lack of a concentration gradient in which the substance has no net movement dynamic equilibrium. While diffusion will go forward in the presence of a substance's concentration gradient, several factors affect the diffusion rate.

  • Extent of the concentration gradient: The greater the difference in concentration, the more rapid the diffusion. The closer the distribution of the material gets to equilibrium, the slower the diffusion rate.
  • Mass of the molecules diffusing: Heavier molecules move more slowly therefore, they diffuse more slowly. The reverse is true for lighter molecules.
  • Temperature: Higher temperatures increase the energy and therefore the molecules' movement, increasing the diffusion rate. Lower temperatures decrease the molecules' energy, thus decreasing the diffusion rate.
  • Solvent density: As the density of a solvent increases, the diffusion rate decreases. The molecules slow down because they have a more difficult time passing through the denser medium. If the medium is less dense, diffusion increases. Because cells primarily use diffusion to move materials within the cytoplasm, any increase in the cytoplasm’s density will inhibit the movement of the materials. An example of this is a person experiencing dehydration. As the body’s cells lose water, the diffusion rate decreases in the cytoplasm, and the cells’ functions deteriorate. Neurons tend to be very sensitive to this effect. Dehydration frequently leads to unconsciousness and possibly coma because of the decrease in diffusion rate within the cells.
  • Solubility: As we discussed earlier, nonpolar or lipid-soluble materials pass through plasma membranes more easily than polar materials, allowing a faster diffusion rate.
  • Surface area and plasma membrane thickness: Increased surface area increases the diffusion rate whereas, a thicker membrane reduces it.
  • Distance travelled: The greater the distance that a substance must travel, the slower the diffusion rate. This places an upper limitation on cell size. A large, spherical cell will die because nutrients or waste cannot reach or leave the cell's center, respectively. Therefore, cells must either be small in size, as in the case of many prokaryotes, or be flattened, as with many single-celled eukaryotes.

A variation of diffusion is the process of filtration. In filtration, material moves according to its concentration gradient through a membrane. Sometimes pressure enhances the diffusion rate, causing the substances to filter more rapidly. This occurs in the kidney, where blood pressure forces large amounts of water and accompanying dissolved substances, or solutes , out of the blood and into the renal tubules. The diffusion rate in this instance is almost totally dependent on pressure. One of the effects of high blood pressure is the appearance of protein in the urine, which abnormally high pressure "squeezes through".

Facilitated transport

In facilitated transport , or facilitated diffusion, materials diffuse across the plasma membrane with the help of membrane proteins. A concentration gradient exists that would allow these materials to diffuse into the cell without expending cellular energy. However, these materials are polar molecule ions that the cell membrane's hydrophobic parts repel. Facilitated transport proteins shield these materials from the membrane's repulsive force, allowing them to diffuse into the cell.

The transported material first attaches to protein or glycoprotein receptors on the plasma membrane's exterior surface. This allows removal of material from the extracellular fluid that the cell needs. The substances then pass to specific integral proteins that facilitate their passage. Some of these integral proteins are collections of beta-pleated sheets that form a pore or channel through the phospholipid bilayer. Others are carrier proteins which bind with the substance and aid its diffusion through the membrane.


The integral proteins involved in facilitated transport are transport proteins , and they function as either channels for the material or carriers. In both cases, they are transmembrane proteins. Channels are specific for the transported substance. Channel proteins have hydrophilic domains exposed to the intracellular and extracellular fluids. In addition, they have a hydrophilic channel through their core that provides a hydrated opening through the membrane layers (Figure 5.9). Passage through the channel allows polar compounds to avoid the plasma membrane's nonpolar central layer that would otherwise slow or prevent their entry into the cell. Aquaporins are channel proteins that allow water to pass through the membrane at a very high rate.

Channel proteins are either open at all times or they are “gated,” which controls the channel's opening. When a particular ion attaches to the channel protein it may control the opening, or other mechanisms or substances may be involved. In some tissues, sodium and chloride ions pass freely through open channels whereas, in other tissues a gate must open to allow passage. An example of this occurs in the kidney, where there are both channel forms in different parts of the renal tubules. Cells involved in transmitting electrical impulses, such as nerve and muscle cells, have gated channels for sodium, potassium, and calcium in their membranes. Opening and closing these channels changes the relative concentrations on opposing sides of the membrane of these ions, resulting in facilitating electrical transmission along membranes (in the case of nerve cells) or in muscle contraction (in the case of muscle cells).

Carrier Proteins

Another type of protein embedded in the plasma membrane is a carrier protein . This aptly named protein binds a substance and, thus triggers a change of its own shape, moving the bound molecule from the cell's outside to its interior (Figure 5.10). Depending on the gradient, the material may move in the opposite direction. Carrier proteins are typically specific for a single substance. This selectivity adds to the plasma membrane's overall selectivity. Scientists poorly understand the exact mechanism for the change of shape. Proteins can change shape when their hydrogen bonds are affected, but this may not fully explain this mechanism. Each carrier protein is specific to one substance, and there are a finite number of these proteins in any membrane. This can cause problems in transporting enough material for the cell to function properly. When all of the proteins are bound to their ligands, they are saturated and the rate of transport is at its maximum. Increasing the concentration gradient at this point will not result in an increased transport rate.

An example of this process occurs in the kidney. In one part, the kidney filters glucose, water, salts, ions, and amino acids that the body requires. This filtrate, which includes glucose, then reabsorbs in another part of the kidney. Because there are only a finite number of carrier proteins for glucose, if more glucose is present than the proteins can handle, the excess is not transported and the body excretes this through urine. In a diabetic individual, the term is “spilling glucose into the urine.” A different group of carrier proteins, glucose transport proteins, or GLUTs, are involved in transporting glucose and other hexose sugars through plasma membranes within the body.

Channel and carrier proteins transport material at different rates. Channel proteins transport much more quickly than carrier proteins. Channel proteins facilitate diffusion at a rate of tens of millions of molecules per second whereas, carrier proteins work at a rate of a thousand to a million molecules per second.


Osmosis is the movement of free water molecules through a semipermeable membrane according to the water's concentration gradient across the membrane, which is inversely proportional to the solutes' concentration. While diffusion transports material across membranes and within cells, osmosis transports only water across a membrane and the membrane limits the solutes' diffusion in the water. Not surprisingly, the aquaporins that facilitate water movement play a large role in osmosis, most prominently in red blood cells and the membranes of kidney tubules.


Osmosis is a special case of diffusion. Water, like other substances, moves from an area of high concentration of free water molecules to one of low free water molecule concentration. An obvious question is what makes water move at all? Imagine a beaker with a semipermeable membrane separating the two sides or halves (Figure 5.11). On both sides of the membrane the water level is the same, but there are different dissolved substance concentrations, or solute , that cannot cross the membrane (otherwise the solute crossing the membrane would balance concentrations on each side). If the solution's volume on both sides of the membrane is the same, but the solute's concentrations are different, then there are different amounts of water, the solvent, on either side of the membrane.

To illustrate this, imagine two full water glasses. One has a single teaspoon of sugar in it whereas, the second one contains one-quarter cup of sugar. If the total volume of the solutions in both cups is the same, which cup contains more water? Because the large sugar amount in the second cup takes up much more space than the teaspoon of sugar in the first cup, the first cup has more water in it.

Returning to the beaker example, recall that it has a solute mixture on either side of the membrane. A principle of diffusion is that the molecules move around and will spread evenly throughout the medium if they can. However, only the material capable of getting through the membrane will diffuse through it. In this example, the solute cannot diffuse through the membrane, but the water can. Water has a concentration gradient in this system. Thus, water will diffuse down its concentration gradient, crossing the membrane to the side where it is less concentrated. This diffusion of water through the membrane—osmosis—will continue until the water's concentration gradient goes to zero or until the water's hydrostatic pressure balances the osmotic pressure. Osmosis proceeds constantly in living systems.


Tonicity describes how an extracellular solution can change a cell's volume by affecting osmosis. A solution's tonicity often directly correlates with the solution's osmolarity. Osmolarity describes the solution's total solute concentration. A solution with low osmolarity has a greater number of water molecules relative to the number of solute particles. A solution with high osmolarity has fewer water molecules with respect to solute particles. In a situation in which a membrane permeable to water, though not to the solute separates two different osmolarities, water will move from the membrane's side with lower osmolarity (and more water) to the side with higher osmolarity (and less water). This effect makes sense if you remember that the solute cannot move across the membrane, and thus the only component in the system that can move—the water—moves along its own concentration gradient. An important distinction that concerns living systems is that osmolarity measures the number of particles (which may be molecules) in a solution. Therefore, a solution that is cloudy with cells may have a lower osmolarity than a solution that is clear, if the second solution contains more dissolved molecules than there are cells.

Hypotonic Solutions

Scientists use three terms—hypotonic, isotonic, and hypertonic—to relate the cell's osmolarity to the extracellular fluid's osmolarity that contains the cells. In a hypotonic situation, the extracellular fluid has lower osmolarity than the fluid inside the cell, and water enters the cell. (In living systems, the point of reference is always the cytoplasm, so the prefix hypo- means that the extracellular fluid has a lower solute concentration, or a lower osmolarity, than the cell cytoplasm.) It also means that the extracellular fluid has a higher water concentration in the solution than does the cell. In this situation, water will follow its concentration gradient and enter the cell.

Hypertonic Solutions

As for a hypertonic solution, the prefix hyper- refers to the extracellular fluid having a higher osmolarity than the cell’s cytoplasm therefore, the fluid contains less water than the cell does. Because the cell has a relatively higher water concentration, water will leave the cell.

Isotonic Solutions

In an isotonic solution, the extracellular fluid has the same osmolarity as the cell. If the cell's osmolarity matches that of the extracellular fluid, there will be no net movement of water into or out of the cell, although water will still move in and out. Blood cells and plant cells in hypertonic, isotonic, and hypotonic solutions take on characteristic appearances (Figure 5.12).

Visual Connection

A doctor injects a patient with what the doctor thinks is an isotonic saline solution. The patient dies, and an autopsy reveals that many red blood cells have been destroyed. Do you think the solution the doctor injected was really isotonic?

Link to Learning

For a video illustrating the diffusion process in solutions, visit this site.

Tonicity in Living Systems

In a hypotonic environment, water enters a cell, and the cell swells. In an isotonic condition, the relative solute and solvent concentrations are equal on both membrane sides. There is no net water movement therefore, there is no change in the cell's size. In a hypertonic solution, water leaves a cell and the cell shrinks. If either the hypo- or hyper- condition goes to excess, the cell’s functions become compromised, and the cell may be destroyed.

A red blood cell will burst, or lyse, when it swells beyond the plasma membrane’s capability to expand. Remember, the membrane resembles a mosaic, with discrete spaces between the molecules comprising it. If the cell swells, and the spaces between the lipids and proteins become too large, the cell will break apart.

In contrast, when excessive water amounts leave a red blood cell, the cell shrinks, or crenates. This has the effect of concentrating the solutes left in the cell, making the cytosol denser and interfering with diffusion within the cell. The cell’s ability to function will be compromised and may also result in the cell's death.

Various living things have ways of controlling the effects of osmosis—a mechanism we call osmoregulation. Some organisms, such as plants, fungi, bacteria, and some protists, have cell walls that surround the plasma membrane and prevent cell lysis in a hypotonic solution. The plasma membrane can only expand to the cell wall's limit, so the cell will not lyse. The cytoplasm in plants is always slightly hypertonic to the cellular environment, and water will always enter a cell if water is available. This water inflow produces turgor pressure, which stiffens the plant's cell walls (Figure 5.13). In nonwoody plants, turgor pressure supports the plant. Conversly, if you do not water the plant, the extracellular fluid will become hypertonic, causing water to leave the cell. In this condition, the cell does not shrink because the cell wall is not flexible. However, the cell membrane detaches from the wall and constricts the cytoplasm. We call this plasmolysis . Plants lose turgor pressure in this condition and wilt (Figure 5.14).

Tonicity is a concern for all living things. For example, paramecia and amoebas, which are protists that lack cell walls, have contractile vacuoles. This vesicle collects excess water from the cell and pumps it out, keeping the cell from lysing as it takes on water from its environment (Figure 5.15).

Many marine invertebrates have internal salt levels matched to their environments, making them isotonic with the water in which they live. Fish, however, must spend approximately five percent of their metabolic energy maintaining osmotic homeostasis. Freshwater fish live in an environment that is hypotonic to their cells. These fish actively take in salt through their gills and excrete diluted urine to rid themselves of excess water. Saltwater fish live in the reverse environment, which is hypertonic to their cells, and they secrete salt through their gills and excrete highly concentrated urine.

In vertebrates, the kidneys regulate the water amount in the body. Osmoreceptors are specialized cells in the brain that monitor solute concentration in the blood. If the solute levels increase beyond a certain range, a hormone releases that slows water loss through the kidney and dilutes the blood to safer levels. Animals also have high albumin concentrations, which the liver produces, in their blood. This protein is too large to pass easily through plasma membranes and is a major factor in controlling the osmotic pressures applied to tissues.


Seminal neuron models, like the McCulloch & Pitts unit [1] or point neurons (see [2] for an overview), assume that synaptic integration is linear. Despite being pervasive mental models of single neuron computation, and frequently used in network models, the linearity assumption has long been known to be false. Measurements using evoked excitatory post-synaptic potentials (EPSPs) have shown that the summation of excitatory inputs can be supra-linear or sub-linear [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], and can summate in quasi-independent regions of dendrite [13].

Supra-linear summation, the dendritic spikes, has been described for a variety of active dendritic mechanisms. For this type of local summation the measured EPSP peak is first above then below the expected arithmetic sum of EPSPs as shown on Figure 1A. Synapse-driven membrane potential depolarization can open [3], [4], [5], [6], or NMDA receptor [6], [7], [8], [9] channels sufficiently to amplify the initial depolarization, and evoke a dendritic spike.

(A–B) The x-axis (Expected EPSP) is the arithmetic sum of two EPSPs induced by two distinct stimulations and y-axis (Measured EPSP) is the measured EPSP when the stimulations are made simultaneously. (A) Observations made on pyramidal neurons (redrawn from [13]). Summation is supra-linear and sub-linear due to the occurrence of a dendritic spike. (B) Â Observations made on cerebellar interneurons (redrawn from [10]). In this case summation is purely sub-linear due to a saturation caused by a reduced driving force. (C) Â The activation function modeling the dendritic spike type non-linear summation: both supra-linear and sub-linear on . (D) Â The activation function modeling the saturation type non-linear summation: strictly sub-linear on . (E) Structure and parameters of the neuron model: and are binary variables describing pre and post-synaptic neuronal activity in circles are two independent sets of non-negative integer-valued synaptic weights respectively for the linear (black) and the non-linear integration (blue) sub-units in the blue square, and are the non-negative integer-valued threshold and height that parameterize the dendritic activation function in the black square is a positive integer-valued threshold determining post-synaptic firing. (F) Â Truth tables of three Boolean functions for inputs: AND, NAND, and XOR. The first column gives the possible values of the input vector the other three columns give the binary outputs in response to each for the three functions considered.

Contrary to the supra-linear summation of dendritic spikes, a saturating sub-linear summation can arise from passive properties of the dendrite [10], [11], [12]. For this type of local summation the measured EPSP peak is always below the expected arithmetic sum of all EPSPs as shown on Figure 1B. Rall's theoretical work [14], [15], subsequently confirmed experimentally [12], showed that passive sub-linear summation of overlapping inputs is a straightforward consequence of the classic model for conductance-driven current injection into the membrane (where , , and are respectively the time varying current, the synaptic conductance, and the membrane voltage, and where is the equilibrium voltage of the channel).

Dendritic spikes inevitably alter the potential range of single neuron computation. Prior theoretical studies found that dendrites could be divided up into multiple, independent sub-units of integration [16], [17], [18], [19], [20] with sigmoidal or Heaviside activation functions (as shown on Figure 1C). They argued that these dendritic spikes turn synaptic integration into a two stage process: first, synaptic inputs are summed in independent sub-units second, the output of these sub-units is linearly summed at the soma. Such a two-stage architecture makes the neuron computationally equivalent to a two-layer artificial neural network, greatly expanding a neuron's computational capacities. It has been shown that spiking dendritic sub-units can enhance the feature storage capacity [18], the generalization capacity [17], [21], the computation of binocular disparity [22], the direction selectivity [23], [24], the creation of multiple place fields [25] or the computation of object-feature binding problems [26]. These enhancements may be explained by the ability of a neuron with a sufficient number of spiking dendritic sub-units to compute linearly non-separable functions whereas seminal neuron models like McCulloch & Pitts cannot [27].

These prior studies made two assumptions that may not generalize to all neurons. Firstly, they supposed that the number of independent dendritic sub-units is potentially large however, for different dendritic morphologies this number may be greatly reduced due to electrotonic coupling or compactness [28], [29], [30]. Secondly, dendritic spikes may not be present in all neuron types, because they lack the specific voltage-gated channels or because the active channel types act to balance each other [10], [11], [31], [32]. Consequently these neuron types could only support a saturating form of non-linear integration. The cerebellar stellate cell is an interesting example because it contradicts both assumptions: it is electrically compact, resulting in a modest number of independent dendritic non-linear sub-units, perhaps on the order of 10 sub-units, as has also been estimated for retinal ganglion cells [33] and its dendrites are passive, with linear integration of inputs in the peri-somatic region and strictly sub-linear integration in the distal dendritic region [10].

If non-linear computation by dendrites were possible for small numbers of sub-units and for passive dendrites, then this would show that enabling linearly non-separable computation by single neurons is, in principle, a general property of dendrites. We thus set out to answer three key questions: (1) whether a single non-linear dendritic sub-unit is sufficient to enable a neuron to compute linearly non-separable functions, so that multiple sub-units are not a necessary requirement (2) whether saturating dendritic sub-units, and not just a spiking dendritic sub-units, are sufficient to enable a neuron to compute linearly non-separable functions and (3) if so, whether the saturating and spiking non-linearities increase computational capacity in the same way.

To answer these questions, we have used a binary neuron model that accounts for non-linear dendritic integration, using either spiking (Figure 1C) or saturating (Figure 1D) activation functions. Using a binary model (Figure 1E) allowed us to study the quantitative increase and qualitative changes in computational capacity using Boolean algebra [34]. A Boolean function is defined by a set of input variables, each taking the value 0 or 1, and a target output value of 0 or 1 for each -dimensional vector that can be made by all possible combinations of values of the input variables (see Material and Methods Boolean Algebra for formal definition). Figure 1F illustrates three well-known examples of Boolean functions, each a function of input variables: AND, NAND, and XOR. The set of Boolean functions computable by a binary neuron model provides a lower bound on the realm of potentially computable algebraic functions by a neuron. Thus, specifying capacity in terms of Boolean functions lets us list the boundaries on a neuron's accessible set of all computable functions.

Using this model, we proceeded on two fronts: first, we used numerical analysis to test if and how much an additional non-linear dendritic sub-unit enables a neuron to compute linearly non-separable functions second, we used formal analytical proofs to show that the numerical results generalise to an arbitrary number of non-linear sub-units. We found numerically that adding a single non-linear dendritic sub-unit, either a spiking or a saturating unit, allows the neuron to compute some positive linearly non-separable Boolean functions. Analytically, we showed that provided a sufficient number of either spiking or saturating dendritic sub-units a neuron is capable of computing all positive linearly non-separable Boolean function.

Second, our numerical analysis showed that a neuron could compute a function using two distinct implementation strategies: a local strategy where each dendritic sub-unit can trigger a somatic spike, implying that the maximal responses of a dendritic sub-unit always correspond to a somatic spike and a global strategy where a somatic spike requires the activation of multiple dendritic sub-units, implying that the maximal response of a dendritic sub-unit may not correspond to a somatic spike. This last result may explain why neurons in layer 2/3 of the visual cortex can stay silent when a calcium response from a dendritic sub-unit is maximal [35]. Analytically, we prove that a neuron with spiking dendritic sub-units can use both strategies to compute a function, whereas a neuron with saturating dendritic sub-units can use only a global strategy to compute a function.

Finally, we show how examples of linearly non-separable functions can be implemented in a reduced, generic biophysical model with either a saturating or a spiking dendrites. Moreover, we show that with electrically compact and passive dendrites, a realistic biophysical model of the cerebellar stellate cell can compute a linearly non-separable function. In conclusion, our study thus extends prior work [16], [19], [20] to show that even a compact neuron with passive dendrites can compute linearly non-separable functions.

Modelling two types of local and non-linear summation

We present in this section the binary neuron models we used to address our questions with a numerical and a formal analysis. We considered two types of dendritic non-linearities modeled by two families of non-linear activation functions , (Figure 1B–C and definition in Materials and Methods spiking and saturating dendritic activation functions). The first family, , modeled dendritic spikes as observed in [6], [13], [4]. The second family, , modeled dendritic saturation as observed in [12], [10], [11]. Both are parameterized by two non-negative parameters for threshold and maximum output .

For our numerical analysis, based on large parameter searches, we added the output of either of these activation functions to a strictly linear sub-unit integrating the same inputs: (1) where is a binary input vector of length , and are non-negative integer-valued weight vectors, and the somatic activation function gives if the result of synaptic integration is above and otherwise. (Note that if is a linear function ( ) then the previous equation can be rewritten as a single linear weighted sum corresponding to the seminal linear neuron model known as the McCulloch & Pitts unit [1]). Poirazi et al [18] have already established that a single non-linear dendritic sub-unit on its own is not sufficient to increase a neuron model's computation capacity. We thus added a non-linear dendritic sub-unit to the somatic non-linearity precisely to assess the impact of adding either a spiking dendritic non-linearity ( ) or a saturating dendritic non-linearity ( ) on single neuron computation. As a corollary, this model includes neuron classes that have a peri-somatic and a distal dendritic region of integration, like cerebellar stellate cell interneurons [10] and layer 2/3 pyramidal neurons [36].

For our formal analysis, which are the three Propositions presented in the Results, we used the generic two-stage neuron model with dendritic sub-units, analogous to [18]: (2) For both our numerical and formal analysis, the neuron input-output mapping is defined as a Boolean function, and each parameter set produces a unique Boolean function . Here we focused on the effect of non-linear EPSP summation, and thus used only non-negative weight vectors. Consequently, an increase in input could only increase (or not change) the output , never decrease it therefore we were studying the neuron's ability to compute positive Boolean functions (see Material and Methods Boolean Algebra for formal definition and Lemma 1 for proof).

In terms of neuron physiology, these binary models are quite general. One can interpret the binary input vector across multiple scales, from the pattern of active and inactive individual pre-synaptic neurons up to the set of active and inactive pre-synaptic cell assemblies afferent to the neuron. In this perspective, the weight represents the peak EPSP magnitude produced when a pre-synaptic neuron or a pre-synaptic cell assembly is active. Similarly the Boolean output of 1 could represent a single spike, a burst of spikes, or a change in rate – whatever it is that is read out by the downstream neurons. For instance, in our biophysical model, a single binary variable corresponds to the synchronous activity of a 100 pre-synaptic neurons, and will show that binary models can lead to informative results in situations where the actual number of pre-synaptic input neurons is in the range consistent with existing data.

Key Points

Neurons receive a plethora of synaptic inputs that are widely spread across their dendritic arbours. This spatial distribution, together with the cable properties of dendrites could cause a pronounced location-dependent variability in the integration properties of synaptic inputs, unless the filtering properties of dendrites are countered.

Although theoretical analyses predict a marked location dependence for the integration of spatially segregated synaptic input, evidence indicates that the dendrites of some cell types can counteract the influence of filtering on the three main elements of synaptic integration — unitary EPSP amplitude, and temporal and spatial summation.

As a result of countering the filtering properties of dendrites, synaptic integration is essentially independent of input location. Three broad categories of cellular properties are involved in the normalization of synaptic integration — neuron morphology, active properties of the membrane and synaptic mechanisms.

Neuron morphology can reduce the location dependence of synaptic integration but its effect cannot completely counteract the filtering effects of dendrites, indicating that the contribution of other cellular properties to linearization is more important.

The active properties of the dendritic membrane are more effective than neuron morphology in reducing the location dependence of integration. Active dendritic conductances endow neurons with the ability to produce both linear and nonlinear interactions between synaptic inputs. However, their exact effect will depend on the specific spatio-temporal characteristics of the input.

Changes in synaptic conductance are the primary mechanism for reducing location dependence of integration in CA1 neurons. The precise nature of the synaptic change is unknown but it is likely to involve an increase in the number of release sites per terminal and increases in the number of synaptic receptors.

The location independence of integration has several functional benefits. For example, it allows a neuron to use Hebbian synaptic mechanisms to store information. Similarly, it could provide a mechanism for the linear encoding of information known to occur in several brain systems by allowing cells that receive spatially divergent patterns of connections from a homogeneous population of neurons to integrate the incoming information as part of the same class of input.

Science Practice Challenge Questions

The capture of radiant energy through the conversion of carbon dioxide and water into carbohydrates is the engine that drives life on Earth. Ribose, C5H10O5, and hexose, C6H12O6, form stable five- and six-carbon rings.

The numbering of the carbons on these rings is important in organizing our description of the role these molecules play in biological energy transfer and information storage and retrieval. Glycolysis is a sequence of chemical reactions that convert glucose to two three-carbon compounds called pyruvic acid.

A. Create visual representations to show how when bonds in the glucose molecules are broken between carbon number 1 and the oxygen atom and between carbons 3 and 4, two molecules of pyruvic acid are produced.

Several enzymes in the cell are involved in converting glucose to pyruvic acid. These enzymes are proteins whose amino acid sequences provide these functions. This protein structure is information that was inherited from the cell’s parent, and is stored in deoxyribonucleic acid (DNA). The “deoxyribo” component of that name is a shorthand for 2-deoxyribose.

B. Create a visual representation of 2-deoxyribose, 5-phosphate by replacing the OH at carbon 2 with a hydrogen atom and replacing the OH at carbon 5 with a hydrogen phosphate ion, HPO3 -2 , whose structure is shown in problem AP3.2. Use your representation to show that both phosphorylation (the addition of a phosphate ion) at carbon 5 and removal of the hydroxide at carbon 2 produce water molecules in an aqueous solution where hydrogen ions are abundant.

DNA is a polymer formed from a chain with repeated 2-deoxyribose, 5-phosphate molecules.

C. Create a visual representation of three 2-deoxyribose, 5-phosphate molecules forming a chain in which an oxygen atom in the phosphate that is attached to the 5-carbon replaces the OH on the 3-carbon of the next ribose sugar.

Cells are bounded by membranes composed of phospholipids. A phospholipid consists of a pair of fatty acids that may or may not have carbon-carbon double bonds, fused at the carboxylic acid with a three-carbon glycerol that is terminated by a phosphate, as shown in the figure below. Most cell membranes comprise two phospholipid layers with the hydrophilic phosphate ends of each molecule in the outer and inner surfaces. The hydrophobic chains of carbon atoms extend into the space between these two surfaces.

The exchange of matter between the interior of the cell and the environment is mediated by this membrane with selective permeability.

A. Pose questions that identify

  • the important characteristics of this lipid bilayer structure
  • the molecules that must be acquired from the environment and eliminated from the cell
  • relationships between the structures of these molecules and the structure of the bilayer

Because the plasma cell membrane has both hydrophilic and hydrophobic properties, few types of molecules possess structures that allow them to pass between the interior of the cell and the environment through passive diffusion. The fluidity of the membrane affects passive transport, and the incorporation of other molecules in the membrane, in particular cholesterols, has a strong effect on its fluidity. Fluidity is also affected by temperature.

Measurements of the speed of movement of oxygen molecules, O2, through three types of membranes were made (Widomska et al., Biochimica et Biophysica Acta, 1,768, 2007) and compared with the speed of movement of O2 through water. These measurements were carried out at four different temperatures. One type of membrane was obtained from the cells in the eyeball of a calf (lens lipid). Synthetic membranes composed of palmitic acid with cholesterol (POPC/CHOL) and without cholesterol (POPC) were also used. The results from these experiments are shown in the table below.

15 25 35 45
Material Speed (cm/s)
Lens lipids 15 30 65 110
POPC/CHOL 15 30 60 95
POPC 55 100 155 280
Water 45 55 65 75

B. Represent these data graphically. The axes should be labeled, and different symbols should be used to plot data for each material.

C. Analyze the data by comparing transport of oxygen through the biological membrane, water, and the synthetic membranes. Consider both membrane composition and temperature in your analysis.

The plasma membrane separates the interior and the exterior of the cell. A potential to do work is established by defining regions inside and outside the cell with different concentrations of key molecules and net charge. In addition to the membrane defining the cell boundary, eukaryotic cells have internal membranes.

D. Explain how internal membranes significantly increase the functional capacity of the cells of eukaryotes relative to those of prokaryotes.

Proteins are polymers whose sub-components are amino acids connected by peptide bonds. The carboxylic acid carbon, O = C – OH, of one amino acid can form a bond with the amine, NH2, of another amino acid. In the formation of this peptide bond, the amine replaces the OH to form O = C – NH2. The other product of this reaction is water, H2O.

Amino acids can be synthesized in the laboratory from simpler molecules of ammonia (NH3), water (H2O), methane (CH4), and hydrogen (H2) if energy is provided by processes that simulate lightning strikes or volcanic eruptions (Miller, Science, 117, 1953 Johnson et al., Science, 322, 2008).

A. The synthesis of amino acids in solutions under laboratory conditions consistent with early Earth was a step toward an explanation of how life began. Pose a question that should have been asked but was not until 2014 (Parker et al., Angewandte Chemie, 53, 2014), when these solutions that had been stored in a refrigerator were analyzed.

The diversity and complexity of life begins in the variety of sequences of the 20 common amino acids.

B. Apply mathematical reasoning to explain the source of biocomplexity by calculating the possible variations in a polymer composed of just three amino acids.

Polarity in a bond between atoms occurs when electrons are distributed unequally. Polarity in a molecule also is caused by charge asymmetry. Life on Earth has evolved within a framework of water, H2O, one of the most polar molecules. The polarities of the amino acids that compose a protein determine the properties of the polymer.

The electric polarity of an amino acid in an aqueous solution depends on the pH of the solution. Here are three forms of the general structure of an amino acid.

C. Qualitatively predict the relationship between solution pH and the form of the amino acid for three solutions of pH: pH < 7, pH = 7, and pH > 7.

The properties of proteins are determined by interactions among the amino acids in the peptide-bonded chain. The protein subcomponents, especially amino R (variable) groups, can interact with very strong charge-charge forces, with attractive forces between groups of atoms with opposite polarities and with repulsive forces between groups of atoms with the same or no polarity. Attractive polar forces often arise between molecules through interactions between oxygen and hydrogen atoms or between nitrogen and hydrogen atoms.

D. Consider particular orientations of pairs of three different amino acids. Predict the relative strength of attractive interaction of all pairs rank them and provide your reasoning.

In an amino acid, the atoms attached to the α carbon are called the R group.

Interactions between R groups of a polypeptide give three-dimensional structure to the one-dimensional, linear sequence of amino acids in a polypeptide.

E. Construct an explanation for the effect of R-group interactions on the properties of a polymer with drawings showing molecular orientations with stronger and weaker polar forces between R groups on asparagine and threonine and between asparagine and alanine.

The nucleobase part of deoxyribonucleic acid encodes information in each component in the sequence making up the polymer. There are five nucleobases that are commonly represented by only a single letter: A (adenine), C (cytosine), G (guanine), T (thymine), and U (uracil). These molecules form a bond with the 1-carbon of deoxyribose. In this problem, we need to look at the molecules in slightly more detail so that you can development the ability to explain why DNA, and sometimes RNA, is the primary source of heritable information.

Edwin Chargaff and his team isolated nucleobases from salmon sperm and determined the fraction of each (Chargaff et al., Journal of Biological Chemistry, 192, 1951). Experiments in which the fraction of all four nucleobases was determined are shown. Also shown are averages as two standard deviations and the sum of total fractions for each experiment. Precision is calculated with each average.

Shown below are the chemical structures of these four nucleobases. In these structures, the nitrogen that attaches to the 2-deoxyribose, 5-phosphate polymer is indicated as N*. The partial charges of particular atoms are indicated with δ + and δ - .

A. Analyze Chargaff’s data in terms of the partial charges on these molecules to show how molecular interactions affect the function of these molecules in the storage and retrieval of biological information.

The interactions between nucleobase molecules are strong enough to produce the association of pairs observed in Chargaff’s data. However, these pairs are bonded by much weaker hydrogen bonds, chemical bonds within the molecules.

Demonstrating an understanding of the replication of DNA requires the ability to explain how the two polymer strands of the double helix interact and grow. To retrieve information from DNA, the strands must be separated. The proteins that perform that task interact with the polymer without forming new chemical bonds. In their paper (Watson and Crick, Nature, 3, 1953) announcing the structure of the polymer that we consider in this problem, Watson and Crick stated, “It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material.”

Eschenmoser and Lowenthal (Chemical Society Reviews, 21, 1992) asked why the 5-carbon sugar ribose is used in DNA when the 6-carbon sugar glucose is so common in biological systems. To answer the question, they synthesized polymeric chains with this alternative form of sugar. They discovered that the strength of the interaction between pairs of nucleobases increased in the new material. Paired strands of hexose-based polymers were more stable.

The AP Biology Curriculum Framework (College Board, 2012) states, “The double-stranded structure of DNA provides a simple and elegant solution for the transmission of heritable information to the next generation by using each strand as a template, existing information can be preserved and duplicated with high fidelity within the replication process. However, the process of replication is imperfect….”

B. Explain why the weaker interaction observed by Eschenmoser and Lowenthal, and the acknowledgement in the Framework that “replication is imperfect,” support the claim implied by Watson and Crick that DNA is the source of heritable information.

Osmolarity vs Molarity

Osmolarity is the number of solute osmoles in one liter of solution. Molarity is the number of solute moles in one liter of a solution. As many substances dissociate in water, it is often more insightful to calculate using osmoles. However, where a solution does not contain dissociated molecules, such as glucose in water, each mole of solute is also one osmol.

The difference between osmolarity and molarity is explained by the van’t Hoff factor – the number of moles (not the mass or weight) of dissociated solute particles (ions) in a solute.

For example, a 1 mol/L glucose solution does not dissociate the van’t Hoff factor is, therefore, one. A solution of 1 mol/L glucose (molarity) has an osmolarity of 1 Osm/L.

However, 1 mol/L solution of calcium chloride (CaCl2) dissociates into three ions. It contains one mole of calcium ions and two moles of chloride ions. The 1 mol/L solution (molarity) multiplied by a van’t Hoff factor of three separate particles means 1 mole CaCl2 is 3 Osm/L.


We investigate the relationship between passive permeability and molecular size, in the context of solubility-diffusion theory, using a diverse compound set with molecular weights ranging from 151 to 828, which have all been characterized in a consistent manner using the RRCK cell monolayer assay. Computationally, each compound was subjected to extensive conformational search and physics-based permeability prediction, and multiple linear regression analyses were subsequently performed to determine, empirically, the relative contributions of hydrophobicity and molecular size to passive permeation in the RRCK assay. Additional analyses of Log D and PAMPA data suggest that these measurements are not size selective, a possible reason for their sometimes weak correlation with cell-based permeability.

Smell and Taste

Thomas Hummel , Johannes Frasnelli , in Handbook of Clinical Neurology , 2019

Temporal summation

The intensity of olfactory stimulants like H2 S increase with a stimulus duration (i.e., show temporal summation ) ( Kobal, 1981 Frasnelli et al., 2006b ) this has also been shown for mixed olfactory–trigeminal stimuli ( Cometto-Muniz and Cain, 1984 ). However, within a mixed stimulus (which comprises most odorous stimuli), the two systems contribute differently to sensations. For example, nicotine produces different trigeminal sensations, including burning and stinging. Using stimuli of 200 ms duration with an interstimulus interval of 2 min, odorous and stinging sensations reached a maximum after 3–5 s while burning does not even commence until 5 s after stimulus onset and does not end until 20 s later ( Hummel et al., 1992b ).

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Triolein-containing standard U.S. Geological Survey (USGS) designed semipermeable membrane devices (SPMDs) were deployed in the field alongside conventional active air sampling equipment for durations of up to 3 months. A high degree of reproducibility between duplicate samples and linear uptake of polychlorinated biphenyls (PCBs) by the USGS SPMDs were observed. USGS SPMD air sampling rates were calculated for a range of PCBs. Sampling rates were found to be higher in winter than in summer and in general increased with increasing chlorination and decreased with increasing ortho-substitution. The sampling rate for the sum of the ICES congeners (IUPAC congeners 28, 52, 101, 118, 138, 153, and 180) was found to be 1.9 m 3 day - 1 SPMD - 1 in summer (mean temperature 18 °C) and 7.6 m 3 day - 1 SPMD - 1 in winter (mean temperature 4 °C). In a separate study USGS SPMDs were deployed for 2 months, and sequestered concentrations and the aforementioned sampling rates were used to calculate atmospheric concentrations. Excellent agreement was found between air concentrations calculated from the SPMDs and active samplers. The immense potential of these lipid-containing USGS SPMDs for time-integrated passive atmospheric monitoring of gas-phase persistent organic pollutants (POPs), for example, in remote areas or for spatial mapping near potential sources, is confirmed.

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Watch the video: Διάλεξη ΜΛ 04 Γραμμικά Υποδείγματα και Άλγεβρα Πινάκων Μέρος 1 (May 2022).


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